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Science Fair Project Encyclopedia The chloride ion is formed when the element chlorine picks up one electron to form the anion (negatively charged ion) Cl−. The salts of hydrochloric acid HCl contain chloride ions and are also called chlorides. An example is table salt, which is sodium chloride with the chemical formula NaCl. In water, it dissolves into Na+ and Cl− ions. The word chloride can also refer to a chemical compound in which one or more chlorine atoms are covalently bonded in the molecule. This means that chlorides can be either inorganic or organic compounds. The simplest example of an inorganic covalently bonded chloride is hydrogen chloride, HCl. A simple example of an organic covalently bonded chloride is chloromethane (CH3Cl), often called methyl chloride. Other examples of inorganic covalently bonded chlorides which are used as reactants are: - phosphorus trichloride, phosphorus pentachloride, and thionyl chloride - all three are reactive chlorinating reagents which have been used in a laboratory. - Disulfur dichloride (SCl2) - used for vulcanization of rubber. Chloride ions have important physiological roles. For instance, in the central nervous system the inhibitory action of glycine and some of the action of GABA relies on the entry of Cl− into specific neurons. The contents of this article is licensed from www.wikipedia.org under the GNU Free Documentation License. Click here to see the transparent copy and copyright details

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Evidence from caves in Siberia indicates that a global temperature increase of 1.5° Celsius may cause substantial thawing of a large tract of permanently frozen soil in Siberia. The thawing of this soil, known as permafrost, could have serious consequences for further changes in the climate. Permafrost regions cover 24 percent of the land surface in the northern hemisphere, and they hold twice as much carbon as is currently present in the atmosphere. As the permafrost thaws, it turns from a carbon sink (meaning it accumulates and stores carbon) into a carbon source, releasing substantial amounts of carbon dioxide and methane into the atmosphere. Both of these gasses enhance the greenhouse effect. By looking at how permafrost has responded to climate change in the past, we can gain a better understanding of climate change today. A team of international researchers looked at speleothems, such as stalagmites, stalactites, and flowstones. These are mineral deposits that are formed when water from snow or rain seeps into the caves. When conditions are too cold or too dry, speleothem growth ceases, since no water flows through the caves. As a result, speleothems provide a detailed history of periods when liquid water was available as well as an assessment of the relationship between global temperature and permafrost extent. Using radioactive dating and data on growth from six Siberian caves, the researchers tracked the history of permafrost in Siberia for the past 450,000 years. The caves were located at varying latitudes, ranging from a boundary of continuous permafrost at 60 degrees North to the permafrost-free Gobi Desert. In the northernmost cave, Lenskaya Ledyanaya, no speleothem growth has occurred since a particularly warm period around 400,000 years ago—the growth at that time suggests water was flowing in the area due to a melt in the permafrost. The extensive thawing at that time allows for an assessment of the warming required globally to cause a similar change in the permafrost boundary. Global temperatures at that time were only 1.5°C warmer than today, suggesting that we could be approaching a critical point at which the coldest permafrost regions would begin to thaw. Not only will increasing global temperatures cause substantial thawing of permafrost, but it may also create wetter conditions in the Gobi Dessert, based on data from the southern-most cave obtained for the same time period. This suggests a dramatically changed environment in continental Asia. Aside from changes in temperature and precipitation, thawing permafrost enables coastal erosion and the liquefaction of ground that was previously frozen. This poses a risk to the infrastructure of Siberia, including major oil and gas facilities.

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Far north within the Arctic Circle off the northern coast of Norway lies a small chain of islands known as Svalbard. These craggy islands have been scoured into shape by ice and sea. The effect of glacial activity can be seen in this image of the northern tip of the island of Spitsbergen. Here, glaciers have carved out a fjord, a U-shaped valley that has been flooded with sea water. Called Bockfjorden, the fjord is located at almost 80 degrees north, and it is still being affected by glaciers. The effect is most obvious in this image in the tan layer of silty freshwater that floats atop the denser blue water of the Arctic Ocean. The fresh water melts off land-bound glaciers and flows over the sandstone, collecting fine red-toned silt. In this image, the tan-colored fresh water flows northward up the fjord and is being pushed to the east side of the fjord by the rotation of the Earth. Glaciers here and elsewhere on Spitsbergen are cold bottom glaciers, which means that they are frozen to the ground rather than floating on top of a thin layer of melt water. The glaciers are also land glaciers since their terminus (end) lies on land, rather than floating on the water (a tidewater glacier). Land glaciers grow and retreat slowly, balancing fresh snow with the melting and draining of old ice. Their rate of growth or retreat can be affected by global warming. In most cases, including the glaciers around Bockfjorden, global warming has caused glaciers to retreat from increased melting. On the eastern side of Svalbard, however, glaciers are growing from enhanced snowfall. The reason for this pattern remains only one of many intriguing unanswered questions of Arctic science in the islands. The Advanced Spaceborne Thermal Emission and Reflection Radiometer, (ASTER) on NASA's Terra satellite captured this false-color image on June 26, 2001.

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5th Grade Oral Language Resources Students will:• Learn about the concept of whales. • Access prior knowledge and build background about whales. • Explore and apply the concept of whales. Students will:• Demonstrate an understanding of the concept of whales. • Orally use words that describe different types of whales and where they live. • Extend oral vocabulary by speaking about terms that describe whales and whale body parts. • Use key concept words [inlet, humpback, ocean, fins, underwater; submerge, ascend, Baleen, mammal]. Explain• Use the slideshow to review the key concept words. • Explain that students are going to learn about: • Where whales live. • Parts of a whale's body. Model• After the host introduces the slideshow, point to the photo on screen. Ask students: What kind of animal do you see in this picture? (whale). What do you know about these animals? (answers will vary). • Ask students: What are the dangers facing whales? (too much hunting, polluted environment). • Say: In this activity, we're going to learn about whales. How can we protect whales? (not pollute the environment, join groups that are concerned with their safety). Guided Practice• Guide students through the next two slides, showing them examples of whales and the way whales live. Always have the students describe how people are different from whales. Apply• Play the games that follow. Have them discuss with their partner the different topics that appear during the Talk About It feature. • After the first game, ask students to talk about what they think a whale's living environment is like. After the second game, have them discuss what they would like and dislike about having the body of a whale. Close• Ask students: How do you move in the water? • Summarize for students that since whales are mammals, they have to come above water to breathe. Encourage them to think about how they breathe underwater.

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A tsunami is a series of waves most commonly caused by violent movement of the sea floor. In some ways, it resembles the ripples radiating outward from the spot where stone has been thrown into the water, but a tsunami can occur on an enormous scale. Tsunamis are generated by any large, impulsive displacement of the sea bed level. The movement at the sea floor leading to tsunami can be produced by earthquakes, landslides and volcanic eruptions. Most tsunamis, including almost all of those traveling across entire ocean basins with destructive force, are caused by submarine faulting associated with large earthquakes. These are produced when a block of the ocean floor is thrust upward, or suddenly drops, or when an inclined area of the seafloor is thrust upward or suddenly thrust sideways. In any event, a huge mass of water is displaced, producing tsunami. Such fault movements are accompanied by earthquakes, which are sometimes referred to as “tsunamigenic earthquakes”. Most tsunamigenic earthquakes take place at the great ocean trenches, where the tectonic plates that make up the earth’s surface collide and are forced under each other. When the plates move gradually or in small thrust, only small earthquakes are produced; however, periodically in certain areas, the plates catch. The overall motion of the plates does not stop; only the motion beneath the trench becomes hung up. Such areas where the plates are hung up are known as “seismic gaps” for their lack of earthquakes. The forces in these gaps continue to build until finally they overcome the strength of the rocks holding back the plate motion. The built-up tension (or comprehension) is released in one large earthquake, instead of many smaller quakes, and these often generate large deadly tsunamis. If the sea floor movement is horizontal, a tsunami is not generated. Earthquakes of magnitude larger than M 6.5 are critical for tsunami generation. Tsunamis produced by landslides: Probably the second most common cause of tsunami is landslide. A tsunami may be generated by a landslide starting out above the sea level and then plunging into the sea, or by a landslide entirely occurring underwater. Landslides occur when slopes or deposits of sediment become too steep and the material falls under the pull of gravity. Once unstable conditions are present, slope failure can be caused by storms, earthquakes, rain, or merely continued deposit of material on the slope. Certain environments are particularly susceptible to the production of landslide-generated earthquakes. River deltas and steep underwater slopes above sub-marine canyons, for instance, are likely sites for landslide-generated earthquakes. Tsunami produced by Volcanoes: The violent geologic activity associated with volcanic eruptions can also generate devastating tsunamis. Although volcanic tsunamis are much less frequent, they are often highly destructive. These may be due to submarine explosions, pyroclastic flows and collapse of volcanic caldera. (1) Submarine volcanic explosions occur when cool seawater encounters hot volcanic magma. It often reacts violently, producing stream explosions. Underwater eruptions at depths of less than 1500 feet are capable of disturbing the water all the way to the surface and producing tsunamis. (2) Pyroclastic flows are incandescent, ground-hugging clouds, driven by gravity and fluidized by hot gases. These flows can move rapidly off an island and into the ocean, their impact displacing sea water and producing a tsunami. (3) The collapse of a volcanic caldera can generate tsunami. This may happen when the magma beneath a volcano is withdrawn back deeper into the earth, and the sudden subsidence of the volcanic edifice displaces water and produces tsunami waves. The large masses of rock that accumulate on the sides of the volcanoes may suddenly slide down slope into the sea, causing tsunamis. Such landslides may be triggered by earthquakes or simple gravitational collapse. A catastrophic volcanic eruption and its ensuing tsunami waves may actually be behind the legend of the lost island civilization of Atlantis. The largest volcanic tsunami in historical times and the most famous historically documented volcanic eruption took lace in the East Indies-the eruption of Krakatau in 1883. Tsunami waves : A tsunami has a much smaller amplitude (wave height) offshore, and a very long wavelength (often hundreds of kilometers long), which is why they generally pass unnoticed at sea, forming only a passing "hump" in the ocean. Tsunamis have been historically referred to tidal waves because as they approach land, they take on the characteristics of a violent onrushing tide rather than the sort of cresting waves that are formed by wind action upon the ocean (with which people are more familiar). Since they are not actually related to tides the term is considered misleading and its usage is discouraged by oceanographers. These waves are different from other wind-generated ocean waves, which rarely extend below a dept of 500 feet even in large storms. Tsunami waves, on the contrary, involvement of water all the way to the sea floor, and as a result their speed is controlled by the depth of the sea. Tsunami waves may travel as fast as 500 miles per hour or more in deep waters of an ocean basin. Yet these fast waves may be only a foot of two high in deep water. These waves have greater wavelengths having long 100 miles between crests. With a height of 2 to 3 feet spread over 100 miles, the slope of even the most powerful tsunamis would be impossible to see from a ship or airplane. A tsunami may consist of 10 or more waves forming a ‘tsunami wave train’. The individual waves follow one behind the other anywhere from 5 to 90 minutes apart. As the waves near shore, they travel progressively more slowly, but the energy lost from decreasing velocity is transformed into increased wavelength. A tsunami wave that was 2 feet high at sea may become a 30-feet giant at the shoreline. Tsunami velocity is dependent on the depth of water through which it travels (velocity equals the square root of water depth h times the gravitational acceleration g, that is (V=√gh). The tsunami will travel approximately at a velocity of 700 kmph in 4000 m depth of sea water. In 10 m, of water depth the velocity drops to about 35 kmph. Even on shore tsunami speed is 35 to 40 km/h, hence much faster than a person can run.It is commonly believed that the water recedes before the first wave of a tsunami crashes ashore. In fact, the first sign of a tsunami is just as likely to be a rise in the water level. Whether the water rises or falls depends on what part of the tsunami wave train first reaches the coast. A wave crest will cause a rise in the water level and a wave trough causes a water recession. Seiche (pronounced as ‘saysh’) is another wave phenomenon that may be produced when a tsunami strikes. The water in any basin will tend to slosh back and forth in a certain period of time determined by the physical size and shape of the basin. This sloshing is known as the seiche. The greater the length of the body, the longer the period of oscillation. The depth of the body also controls the period of oscillations, with greater water depths producing shorter periods. A tsunami wave may set off seiche and if the following tsunami wave arrives with the next natural oscillation of the seiche, water may even reach greater heights than it would have from the tsunami waves alone. Much of the great height of tsunami waves in bays may be explained by this constructive combination of a seiche wave and a tsunami wave arriving simultaneously. Once the water in the bay is set in motion, the resonance may further increase the size of the waves. The dying of the oscillations, or damping, occurs slowly as gravity gradually flattens the surface of the water and as friction turns the back and forth sloshing motion into turbulence. Bodies of water with steep, rocky sides are often the most seiche-prone, but any bay or harbour that is connected to offshore waters can be perturbed to form seiche, as can shelf waters that are directly exposed to the open sea. The presence of a well developed fringing or barrier of coral reef off a shoreline also appears to have a strong effect on tsunami waves. A reef may serve to absorb a significant amount of the wave energy, reducing the height and intensity of the wave impact on the shoreline itself. The popular image of a tsunami wave approaching shore is that of a nearly vertical wall of water, similar to the front of a breaking wave in the surf. Actually, most tsunamis probably don’t form such wave fronts; the water surface instead is very close to the horizontal, and the surface itself moves up and down. However, under certain circumstances an arriving tsunami wave can develop an abrupt steep front that will move inland at high speeds. This phenomenon is known as a bore. In general, the way a bore is created is related to the velocity of the shallow water waves. As waves move into progressively shallower water, the wave in front will be traveling more slowly than the wave behind it .This phenomenon causes the waves to begin “catching up” with each other, decreasing their distance apart i.e. shrinking the wavelength. If the wavelength decreases, but the height does not, then waves must become steeper. Furthermore, because the crest of each wave is in deeper water than the adjacent trough, the crest begins to overtake the trough in front and the wave gets steeper yet. Ultimately the crest may begin to break into the trough and a bore formed. A tsunami can cause a bore to move up a river that does not normally have one. Bores are particularly common late in the tsunami sequence, when return flow from one wave slows the next incoming wave. Though some tsunami waves do, in deed, form bores, and the impact of a moving wall of water is certainly impressive, more often the waves arrive like a very rapidly rising tide that just keeps coming and coming. The normal wind waves and swells may actually ride on top of the tsunami, causing yet more turbulence and bringing the water level to even greater heights.

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Learn something new every day More Info... by email A predicate is part of a sentence or clause in English and is one of two primary components that serves to effectively complete the sentence. Sentences consist of two main components: subjects and predicates. Subjects are the primary “thing” in a sentence which the rest of the words then describe through either a direct description or by indicating what type of action that subject is performing. The predicate is this secondary aspect of the sentence and usually consists of a verb or adjective, though complicated sentences may have multiple verbs and a number of descriptions affecting the subject. It can be easiest to understand predicates by first understanding subjects and how sentences are constructed. A sentence just about always has a subject, though it can be implied in some way and not necessarily directly stated. In a simple sentence like “The cat slept,” the subject is “the cat,” which is a noun phrase consisting of the direct article “the” and the noun “cat.” Subjects can be longer and more complicated, but they are usually fairly simple in nature. The predicate of a sentence is then basically the rest of the sentence, though this is not always the case for longer and more complicated sentences. In “The cat slept,” the predicate is quite simple and merely consists of the word “slept.” This is simple because “slept” is an intransitive verb, which means that it requires no further description or objects to make it complete. The sentence could be expanded as “The cat slept on the bed,” but this is not necessary and merely adds a descriptive component to the predicate through the prepositional phrase “on the bed.” In a somewhat more complicated sentence, such as “The man gave the ball to his son,” the subject of the sentence is still quite simple: “The man.” The predicate in this sentence, however, has become substantially more complicated and consists of the rest of the sentence: “gave the ball to his son.” This has been made more complicated because the verb “gave” is transitive, specifically ditransitive, which indicates both a direct object and an indirect object. The act of “giving” requires that there is a direct object, which is the item given, and an indirect object, which is who or what it is given to. In this instance, the predicate consists of the verb “gave” and the direct object “the ball” with a connecting preposition “to” and the indirect object “his son.” Predicates can become even more complicated as an idea expands, such as a sentence like “The rock rolled off the table, landed on top of a skateboard, and proceeded to roll down the hill until it was stopped by a wall.” In this sentence, the subject is only “The rock,” which means that the rest of the sentence is the predicate.

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Prefixes, Suffixes, Inflectional Endings, and Root Words Spelling Words Correctly Using Prefixes This strategy will focus on the prefix "re-" to help predict the meaning of words. The same strategy can be used to introduce other common prefixes such as "dis-", "in-" and "im-". Prefixes and Suffixes Students will create two Mini Books. One will incorporate prefixes and the other will focus on suffixes. Each book will include the meanings, sample words, and two well- written sentences for each suffix or prefix. Also supports Tech COS 12 Making Singular Nouns Plural This lesson involves the use of the Structural Analysis element of the Inflectional Ending "-s" to make singular nouns plural. Vocabulary Root Word Drawing A Lesson Plans Page lesson plan, lesson idea, thematic unit, or activity in Language Arts and Art called Vocabulary Root Word Drawing. Forming Possessives Showing possession in English is a relatively easy matter (believe it or not). By adding an apostrophe and an s we can manage to transform most singular nouns into their possessive form: Word Confusion: Students choose the correct word to complete the sentence in this online game. Inflected Endings: Some languages, such as Chinese, Hmong, and Vietnamese do not use inflected endings to form verb tenses. Students may need help understanding that adding -ed to a verb indicates that the action happened in the past. Spelling changes in inflected verbs may be difficult for ELLs to master. Prefixes and Suffixes: Some English prefixes and suffixes have equivalent forms in the Romance languages. For example, the prefix dis- in English (disapprove) corresponds to the Spanish des- (desaprobar), the French des- (desapprouver), and the Haitian Creole dis- or dez- (dezaprouve). Students who are literate in a Romance language may be able to transfer their understanding of prefixes and suffixes much easier than those from non-Romance languages. E/B, D, E: Help ELLs classify English words into meaningful categories. Use word walls, graphic organizers, and concept maps to group related words, record them in meaningful ways, and create visual references that can be used in future lessons. Teachers can help students group and relate words in different ways. For example, place a large picture of a tree on the wall. Place prefix and suffix cards on the different branches (i.e. prefixes: pre-, re- un-; suffixes: -ful, -less) and root words on the roots (write, view, paint). This visual representation can help students conceptualize that prefixes and suffixes are added on to root words. E/B, D, E: The teacher creates a display of words containing Greek and Latin roots and adds to it during the school year. ELLs can refer to the display to help in understanding new words. (Example of display: the tree display above, or a poster with three columns - Root, Meaning, and Word, i.e. aqua, water, aquarium) E/B: Read one's own writing or simple narrative text and begin to produce phonemes appropriately. E/B: Recognize and produce English phonemes students already know, and possibly use them in simple phrases or sentences. E/B: Recognize sounds in spoken words with accompanying illustrations E/B: Use cues for sounding out unfamiliar words with accompanying illustrations E/B: Blend sounds together to make words, shown visually D: Remove or add sounds to existing words to make new words, shown visually (i.e. "Cover up the t in cart. What do you have now?") D: Use letter-sound relationships and word roots to produce and understand multi-syllabic words; E: Use letter-sound relationships and word roots to produce and understand new word families. D, E: Recognize and use prefixes and suffixes to find meanings of unknown words. E: Segment illustrated sentences into words and phrases. E: Identify and analyze sentence and context clues to find meanings of unknown words. E/B, D, E: When sharing new vocabulary words, make sure to write each word divided into syllables (i.e. dic-tion-ar-y). When introducing each word, sound it out, pausing between each syllable, and then blend the syllables together. Have students repeat after you. Ask students how many syllables the word has. Tell students: Pay attention to the syllables in a word. This will help you spell the word, and it will help you pronounce it, too. E/B, D, E: Before teaching the phonics skills, introduce the target words orally to students by using them in activities such as chants and riddle games, or asking and answering questions that use the words. Some of the above ELL suggestions came from the following resources: WIDA Consortium's English Language Proficiency Standards and Resource Guide, PreK - Grade 12 Scott Foresman Reading Street ELL and Transition Handbook Grades 3-6 A Guide to the Standard Course of Study for Limited English Proficient Students / Grades K-5 (Public Schools of N.C.)

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Say it with FEELING!! Rationale: “Fluency means reading faster, smoother, more expressively, or more quietly with the goal of reading silently. Fluent reading approaches the speed of speech.” (Murray) At this development stage, fluency is a major goal of the student and the teacher. This lesson is aimed to teach and emphasize one aspect of fluency: expression. Reading with expression brings a story, and its characters, to life, making reading more enjoyable for everyone. The teacher will read a story, showing great expression, to model for children. Materials: Copy of Tiki Tiki Tembo, various classroom library books, notebook paper, pencils 1. Review with students the difference that punctuation makes make at the end of a sentence. Read the following sentences twice through. The first time, pay NO ATTENTION to the punctuation marks at the end of the sentence. The second time, use the correct inflection in your voice, depending on the punctuation mark at the end of the sentence. “JIMMY WENT RUNNING., JIMMY WENT RUNNING?, JIMMY WENT RUNNING!. CAN ANYONE TELL ME THE DIFFERENCES IN THOSE SENTENCES?” Hopefully children will answer that the first was a statement, the second was a question, and the third was an exclamation. 2. “WHAT A WONDERFUL DAY WE HAVE!!!” After you have excited the kids with that exclamation, the teacher says ‘“NOW THAT WAS LOUD AND FULL OF EXCITEMENT WASN’T IT? THAT WAS HAPPY EXPRESSION. WHEN WE TALK OR READ WITH EXPRESSION, WE CHANGE THE TONE OF OUR VOICE (HAPPY TO SAD), THE VLOUME OF OUR VOICE (LOUD TO SOFT), AND USE OUR FACES TO SHOW THE FEELING OF THE BOOK. DIFFERENT FEELINGS HAVE DIFFERENT SOUNDS AND FACIL LOOKS.” 3. “CAN SOMEONE TELL ME WHY WE SHOULD USE EXPRESSION WHEN WE READ? Students will offer their own explanations. “GREAT! WE USE EXPRESSION TO MAKE THE STORY MORE INTERESTING AND FUN TO READ!!!” 4. “WHAT WOULD MY VOICE SOUND LIKE IF I WERE SCARED?” Children raise their hands and answer, using facial expressions and vocal tones. “WHAT ABOUT IF I WERE ANGRY? WOULD I YELL OR WHISPER?” Children will answer correctly to the question. 5. Now, gather the children around your reading center and read ‘“Tiki Tiki Tembo’”. Make sure to OVEREXAGGERATE your expressions. (vocal tone, facial expressions, and volume) When done reading, ask children what emotions you were trying to convey at different parts of the story. Have a mini group discussion. 6. Pair children up and have them select a book from the classroom library to read. Set a timer for 5-8 minutes and let each child read to their partner. “REMEMEBER TO READ TO YOUR READING BUDDY WITH LOTS OF EXPRESSION! MAKE YOUR READING BUDDY FEEL LIKE THEY ARE IN THE STORY.” Teacher circulates with rubric and evaluates each child as they read. Now have the kids switch roles. Reading buddy becomes reader and reader becomes reading buddy. 7. After the children are done with the reading, have each child individually write three sentences about their book that end with various punctuation marks. “OKAY CLASS, NOW THAT WE HAVE LEARNED TO READ WITH EXPRESSION, I WANT US TO WRITE WITH EXPRESSION. TAKE OUT PAPER AND A PENCIL. WRITE THREE SENTENCES ABOUT THE STORY YOU JUST READ. ONE SHOULD BE A STATEMENT AND END WITH A PERIOD. ONE SHOULD BE A QUESTION AND END WITH A QUESTION MARK. ONE SHOULD BE AN EXCLAMATION AND END WITH AN EXCLAMATION POINT.” Have each child come to your desk or reading table and have them read, with expression, their original sentences. This will assess their grasp of punctuation and also the concept of expression: how to write it and convey it to the reader. You also have the checklist rubric that you evaluated their oral reading on. www.auburn.edu/rdggenie The Reading Genie Website Click here to return to Openings

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What is the media? What does it do? Students examine the types and roles of the media by taking on the role of newsmaker and agenda setter. Students will be able to: ANTICIPATE by asking students if they’ve ever seen a television newscast. Ask students to recall any details they remember (graphics, music, story topics). Ask students who they think makes decisions about what stories television newscasts discuss. DISTRIBUTE the Reading pages to each student. READ the two reading pages with the class, pausing to discuss as necessary. CHECK for understanding by doing the T/F Active Participation activity. Have students respond “True” or “False” as a chorus or use thumbs up/thumbs down. DISTRIBUTE scissors, glue, and the Agenda Cutout Activity pages. Students can complete this activity individually or in pairs. READ the directions for the cutout activity. ALLOW students to complete the cutout activity. REVIEW the answers to the cutout activity. DISTRIBUTE one worksheet to each student and review the directions for the activities. ALLOW students to complete the worksheet. DISTRIBUTE one Extension Activity to each student and review the directions. ALLOW students to complete the extension activity. CLOSE by asking students to silently recall as many roles of the media as they can. Call on students until all roles have been named.

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You must be familiar with slope-intercept form (y = mx + b), and understand which numbers in the equation are m and b, and how to graph them. Mark b on the graph, then graph the slope (m) from that point. Inequalities are very similar, with only a few differences: - It's not a line of solutions as in a linear equation; it is a solid or dashed boundary line that shows on which side all the solutions are. - Shade above or below the boundary line, showing on which side all the solutions are. - Change the direction of the inequality (>, <) if you divide by a negative number. ►Check your work by using (0,0) as a test point. This will help you know if your answer is correct, and if you forgot to change the direction of the inequality. These videos cover the same topic, but go about solving in slightly different ways. I watched all of them, and gleaned a little more from each one. (1) from YourTeacher.com - graphing using a table (2) boundary line (3) graphing using slope-intercept form, y = mx + b (4) graphing using slope-intercept form. He is fast, so pause and read the text on the board. (5) graphing using slope-intercept form

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Delegates (C# Programming Guide) A delegate is a type that defines a method signature. When you instantiate a delegate, you can associate its instance with any method with a compatible signature. You can invoke (or call) the method through the delegate instance. Delegates are used to pass methods as arguments to other methods. Event handlers are nothing more than methods that are invoked through delegates. You create a custom method, and a class such as a windows control can call your method when a certain event occurs. The following example shows a delegate declaration: Any method from any accessible class or struct that matches the delegate's signature, which consists of the return type and parameters, can be assigned to the delegate. The method can be either static or an instance method. This makes it possible to programmatically change method calls, and also plug new code into existing classes. As long as you know the signature of the delegate, you can assign your own method. In the context of method overloading, the signature of a method does not include the return value. But in the context of delegates, the signature does include the return value. In other words, a method must have the same return value as the delegate. This ability to refer to a method as a parameter makes delegates ideal for defining callback methods. For example, a reference to a method that compares two objects could be passed as an argument to a sort algorithm. Because the comparison code is in a separate procedure, the sort algorithm can be written in a more general way. Delegates have the following properties: Delegates are like C++ function pointers but are type safe. Delegates allow methods to be passed as parameters. Delegates can be used to define callback methods. Delegates can be chained together; for example, multiple methods can be called on a single event. Methods do not have to match the delegate signature exactly. For more information, see Using Variance in Delegates (C# and Visual Basic). C# version 2.0 introduced the concept of Anonymous Methods, which allow code blocks to be passed as parameters in place of a separately defined method. C# 3.0 introduced lambda expressions as a more concise way of writing inline code blocks. Both anonymous methods and lambda expressions (in certain contexts) are compiled to delegate types. Together, these features are now known as anonymous functions. For more information about lambda expressions, see Anonymous Functions (C# Programming Guide). For more information, see the C# Language Specification. The language specification is the definitive source for C# syntax and usage.

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The people in south Asia had no warning of the next disaster rushing toward them the morning of December 26, 2004. One of the strongest earthquakes in the past 100 years had just destroyed villages on the island of Sumatra in the Indian Ocean, leaving many people injured. But the worst was yet to come—and very soon. For the earthquake had occurred beneath the ocean, thrusting the ocean floor upward nearly 60 feet. The sudden release of energy into the ocean created a tsunami (pronounced su-NAM-ee) event—a series of huge waves. The waves rushed outward from the center of the earthquake, traveling around 400 miles per hour. Anything in the path of these giant surges of water, such as islands or coastlines, would soon be under water. The people had already felt the earthquake, so why didn't they know the water was coming? As the ocean floor rises near a landmass, it pushes the wave higher. But much depends on how sharply the ocean bottom changes and from which direction the wave approaches. Energy from earthquakes travels through the Earth very quickly, so scientists thousands of miles away knew there had been a severe earthquake in the Indian Ocean. Why didn't they know it would create a tsunami? Why didn't they warn people close to the coastlines to get to higher ground as quickly as possible? In Sumatra, near the center of the earthquake, people would not have had time to get out of the way even if they had been warned. But the tsunami took over two hours to reach the island of Sri Lanka 1000 miles away, and still it killed 30,000 people! It is important, though, to understand just how the tsunami will behave when it gets near the coastline. As the ocean floor rises near a landmass, it pushes the wave higher. But much depends on how sharply the ocean bottom changes and from which direction the wave approaches. Scientists would like to know more about how actual waves react. MISR has nine cameras all pointed at different angles. So the exact same spot is photographed from nine different angles as the satellite passes overhead. The image at the top of this page was taken with the camera that points forward at 46°. The image caught the sunlight reflecting off the pattern of ripples as the waves bent around the southern tip of the island. These ripples are not seen in satellite images looking straight down at the surface. Scientists do not yet understand what causes this pattern of ripples. They will use computers to help them find out how the depth of the ocean floor affects the wave patterns on the surface of the ocean. Images such as this one from MISR will help. Images such as these from MISR will help scientists understand how tsunamis interact with islands and coastlines. This information will help in developing the computer programs, called models, that will help predict where, when, and how severely a tsunami will hit. That way, scientists and government officials can warn people in time to save many lives.

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Simple Equations Introduction to basic algebraic equations of the form Ax=B ⇐ Use this menu to view and help create subtitles for this video in many different languages. You'll probably want to hide YouTube's captions if using these subtitles. - Let's say we have the equation seven times x is equal to fourteen. - Now before even trying to solve this equation, - what I want to do is think a little bit about what this actually means. - Seven x equals fourteen, - this is the exact same thing as saying seven times x, let me write it this way, seven times x, x in orange again. Seven times x is equal to fourteen. - Now you might be able to do this in your head. - You could literally go through the 7 times table. - You say well 7 times 1 is equal to 7, so that won't work. - 7 times 2 is equal to 14, so 2 works here. - So you would immediately be able to solve it. - You would immediately, just by trying different numbers - out, say hey, that's going to be a 2. - But what we're going to do in this video is to think about - how to solve this systematically. - Because what we're going to find is as these equations get - more and more complicated, you're not going to be able to - just think about it and do it in your head. - So it's really important that one, you understand how to - manipulate these equations, but even more important to - understand what they actually represent. - This literally just says 7 times x is equal to 14. - In algebra we don't write the times there. - When you write two numbers next to each other or a number next - to a variable like this, it just means that you - are multiplying. - It's just a shorthand, a shorthand notation. - And in general we don't use the multiplication sign because - it's confusing, because x is the most common variable - used in algebra. - And if I were to write 7 times x is equal to 14, if I write my - times sign or my x a little bit strange, it might look - like xx or times times. - So in general when you're dealing with equations, - especially when one of the variables is an x, you - wouldn't use the traditional multiplication sign. - You might use something like this -- you might use dot to - represent multiplication. - So you might have 7 times x is equal to 14. - But this is still a little unusual. - If you have something multiplying by a variable - you'll just write 7x. - That literally means 7 times x. - Now, to understand how you can manipulate this equation to - solve it, let's visualize this. - So 7 times x, what is that? - That's the same thing -- so I'm just going to re-write this - equation, but I'm going to re-write it in visual form. - So 7 times x. - So that literally means x added to itself 7 times. - That's the definition of multiplication. - So it's literally x plus x plus x plus x plus x -- let's see, - that's 5 x's -- plus x plus x. - So that right there is literally 7 x's. - This is 7x right there. - Let me re-write it down. - This right here is 7x. - Now this equation tells us that 7x is equal to 14. - So just saying that this is equal to 14. - Let me draw 14 objects here. - So let's say I have 1, 2, 3, 4, 5, 6, 7, 8, - 9, 10, 11, 12, 13, 14. - So literally we're saying 7x is equal to 14 things. - These are equivalent statements. - Now the reason why I drew it out this way is so that - you really understand what we're going to do when we - divide both sides by 7. - So let me erase this right here. - So the standard step whenever -- I didn't want to do that, - let me do this, let me draw that last circle. - So in general, whenever you simplify an equation down to a - -- a coefficient is just the number multiplying - the variable. - So some number multiplying the variable or we could call that - the coefficient times a variable equal to - something else. - What you want to do is just divide both sides by 7 in - this case, or divide both sides by the coefficient. - So if you divide both sides by 7, what do you get? - 7 times something divided by 7 is just going to be - that original something. - 7's cancel out and 14 divided by 7 is 2. - So your solution is going to be x is equal to 2. - But just to make it very tangible in your head, what's - going on here is when we're dividing both sides of the - equation by 7, we're literally dividing both sides by 7. - This is an equation. - It's saying that this is equal to that. - Anything I do to the left hand side I have to do to the right. - If they start off being equal, I can't just do an operation - to one side and have it still be equal. - They were the same thing. - So if I divide the left hand side by 7, so let me divide - it into seven groups. - So there are seven x's here, so that's one, two, three, - four, five, six, seven. - So it's one, two, three, four, five, six, seven groups. - Now if I divide that into seven groups, I'll also want - to divide the right hand side into seven groups. - One, two, three, four, five, six, seven. - So if this whole thing is equal to this whole thing, then each - of these little chunks that we broke into, these seven chunks, - are going to be equivalent. - So this chunk you could say is equal to that chunk. - This chunk is equal to this chunk -- they're - all equivalent chunks. - There are seven chunks here, seven chunks here. - So each x must be equal to two of these objects. - So we get x is equal to, in this case -- in this case - we had the objects drawn out where there's two of - them. x is equal to 2. - Now, let's just do a couple more examples here just so it - really gets in your mind that we're dealing with an equation, - and any operation that you do on one side of the equation - you should do to the other. - So let me scroll down a little bit. - So let's say I have I say I have 3x is equal to 15. - Now once again, you might be able to do is in your head. - You're saying this is saying 3 times some - number is equal to 15. - You could go through your 3 times tables and figure it out. - But if you just wanted to do this systematically, and it - is good to understand it systematically, say OK, this - thing on the left is equal to this thing on the right. - What do I have to do to this thing on the left - to have just an x there? - Well to have just an x there, I want to divide it by 3. - And my whole motivation for doing that is that 3 times - something divided by 3, the 3's will cancel out and I'm just - going to be left with an x. - Now, 3x was equal to 15. - If I'm dividing the left side by 3, in order for the equality - to still hold, I also have to divide the right side by 3. - Now what does that give us? - Well the left hand side, we're just going to be left with - an x, so it's just going to be an x. - And then the right hand side, what is 15 divided by 3? - Well it is just 5. - Now you could also done this equation in a slightly - different way, although they are really equivalent. - If I start with 3x is equal to 15, you might say hey, Sal, - instead of dividing by 3, I could also get rid of this 3, I - could just be left with an x if I multiply both sides of - this equation by 1/3. - So if I multiply both sides of this equation by 1/3 - that should also work. - You say look, 1/3 of 3 is 1. - When you just multiply this part right here, 1/3 times - 3, that is just 1, 1x. - 1x is equal to 15 times 1/3 third is equal to 5. - And 1 times x is the same thing as just x, so this is the same - thing as x is equal to 5. - And these are actually equivalent ways of doing it. - If you divide both sides by 3, that is equivalent to - multiplying both sides of the equation by 1/3. - Now let's do one more and I'm going to make it a little - bit more complicated. - And I'm going to change the variable a little bit. - So let's say I have 2y plus 4y is equal to 18. - Now all of a sudden it's a little harder to - do it in your head. - We're saying 2 times something plus 4 times that same - something is going to be equal to 18. - So it's harder to think about what number that is. - You could try them. - Say if y was 1, it'd be 2 times 1 plus 4 times 1, - well that doesn't work. - But let's think about how to do it systematically. - You could keep guessing and you might eventually get - the answer, but how do you do this systematically. - Let's visualize it. - So if I have two y's, what does that mean? - It literally means I have two y's added to each other. - So it's literally y plus y. - And then to that I'm adding four y's. - To that I'm adding four y's, which are literally four - y's added to each other. - So it's y plus y plus y plus y. - And that has got to be equal to 18. - So that is equal to 18. - Now, how many y's do I have here on the left hand side? - How many y's do I have? - I have one, two, three, four, five, six y's. - So you could simplify this as 6y is equal to 18. - And if you think about it it makes complete sense. - So this thing right here, the 2y plus the 4y is 6y. - So 2y plus 4y is 6y, which makes sense. - If I have 2 apples plus 4 apples, I'm going - to have 6 apples. - If I have 2 y's plus 4 y's I'm going to have 6 y's. - Now that's going to be equal to 18. - And now, hopefully, we understand how to do this. - If I have 6 times something is equal to 18, if I divide both - sides of this equation by 6, I'll solve for the something. - So divide the left hand side by 6, and divide the - right hand side by 6. - And we are left with y is equal to 3. - And you could try it out. - That's what's cool about an equation. - You can always check to see if you got the right answer. - Let's see if that works. - 2 times 3 plus 4 times 3 is equal to what? - 2 times 3, this right here is 6. - And then 4 times 3 is 12. - 6 plus 12 is, indeed, equal to 18. Be specific, and indicate a time in the video: At 5:31, how is the moon large enough to block the sun? Isn't the sun way larger? Have something that's not a question about this content? This discussion area is not meant for answering homework questions. Share a tip When naming a variable, it is okay to use most letters, but some are reserved, like 'e', which represents the value 2.7831... Have something that's not a tip or feedback about this content? This discussion area is not meant for answering homework questions. Discuss the site For general discussions about Khan Academy, visit our Reddit discussion page. Flag inappropriate posts Here are posts to avoid making. 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Plot Cartesian Coordinate Points on a Cartesian Graph When math folks talk about using a graph, they’re usually referring to a Cartesian graph (also called the Cartesian coordinate system). The below figure shows an example of a Cartesian graph. A Cartesian graph is really just two number lines that cross at 0. These number lines are called the horizontal axis (also called the x-axis) and the vertical axis (also called the y-axis). The place where these two axes (plural of axis) cross is called the origin. Plotting a point (finding and marking its location) on a graph isn’t much harder than finding a point on a number line, because a graph is just two number lines put together. Every point on a Cartesian graph is represented by two numbers in parentheses, separated by a comma, called a set of coordinates. To plot any point, start at the origin, where the two axes cross. The first number tells you how far to go to the right (if positive) or left (if negative) along the horizontal axis. The second number tells you how far to go up (if positive) or down (if negative) along the vertical axis. For example, here are the coordinates of four points called A, B, C, and D: A = (2, 3) B = (–4, 1) C = (0, –5) D = (6, 0) The above figure depicts a graph with these four points plotted. Start at the origin, (0, 0). To plot point A, count 2 spaces to the right and 3 spaces up. To plot point B, count 4 spaces to the left (the negative direction) and then 1 space up. To plot point C, count 0 spaces left or right and then count 5 spaces down (the negative direction). And to plot point D, count 6 spaces to the right and then 0 spaces up or down.

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Lesson Plans for Secondary School Educators Unit Nine: "The Quest Is Achieved" Content Focus: The Lord of the Rings, Book Six Thematic Focus: What Makes a Hero? As befits its vast scope and extraordinary ambition, The Lord of the Rings boasts three major heroes Frodo, Aragorn, and Sam plus many secondary characters whose deeds are manifestly noble and courageous. In Unit Nine students consider the meaning of heroism and look back on the other thematic threads that make the novel a unified whole. By the end of Unit Nine, the student should be able to: Contrast Aragorn's obvious valor with Frodo's concealed heroism. Give some possible reasons Tolkien regarded Sam as the "chief hero" of The Lord of the Rings. Account for the "joy-in-sorrow atmosphere" of Tolkien's epic fantasy. Indicate which of Tolkien's characters might be considered archetypes. Trace the development of the novel's themes, including corruption, free will, destiny, despair, and heroism, from Book One through Book Six. Unit Nine Content Comments for Teachers These lesson plans were written by James Morrow and Kathryn Morrow in consultation with Amy Allison, Gregory Miller, Sarah Rito, and Jason Zanitsch. Lesson Plans Homepage

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The OSI (Open Systems Interconnection) model was created by the ISO to help standardize communication between computer systems. It divides communications into seven different layers, which each include multiple hardware standards, protocols, or other types of services. The seven layers of the OSI model include: - The Physical layer - The Data Link layer - The Network layer - The Transport layer - The Session layer - The Presentation layer - The Application layer When one computer system communicates with another, whether it is over a local network or the Internet, data travels through these seven layers. It begins with the physical layer of the transmitting system and travels through the other layers to the application layer. Once the data reaches the application layer, it is processed by the receiving system. In some cases, the data will move through the layers in reverse to the physical layer of the receiving computer. The best way to explain how the OSI model works is to use a real life example. In the following illustration, a computer is using a wireless connection to access a secure website. The communications stack begins with the (1) physical layer. This may be the computer's Wi-Fi card, which transmits data using the IEEE 802.11n standard. Next, the (2) data link layer might involve connecting to a router via DHCP. This would provide the system with an IP address, which is part of the (3) network layer. Once the computer has an IP address, it can connect to the Internet via the TCP protocol, which is the (4) transport layer. The system may then establish a NetBIOS session, which creates the (5) session layer. If a secure connection is established, the (6) presentation layer may involve an SSL connection. Finally, the (7) application layer consists of the HTTP connection to the website. The OSI model provides a helpful overview of the way computer systems communicate with each other. Software developers often use this model when writing software that requires networking or Internet support. Instead of recreating the communications stack from scratch, software developers only need to include functions for the specific OSI layer(s) their programs use.

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On this day in 1784, at the Maryland State House in Annapolis, the Continental Congress ratifies the Treaty of Paris. The document, negotiated in part by future President John Adams, contained terms for ending the Revolutionary War and established the United States as a sovereign nation. The treaty outlined America's fishing rights off the coast of Canada, defined territorial boundaries in North America formerly held by the British and forced an end to reprisals against British loyalists. Two other future presidents, Thomas Jefferson and James Monroe, were among the delegates who ratified the document on January 14, 1874. Thomas Jefferson had planned to travel to Paris to join Adams, John Jay and Benjamin Franklin for the beginning of talks with the British in 1782. However, after a delay in his travel plans, Jefferson received word that a cessation of hostilities had been announced by King George III the previous December. Jefferson arrived in Paris in late February after the treaty had already been negotiated by Adams, Franklin and Jay. Adams' experience and skill in diplomacy prompted Congress to authorize him to act as the United States' representative in negotiating treaty terms with the British. Following his role in ending the Revolutionary War and his participation in drafting the Declaration of Independence, Adams succeeded George Washington as the second president of the United States in 1797.

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The process that breaks up and carries away the rocks and soils that make up the Earth’s surface is called erosion. It is caused by flowing water, waves, glaciers, and the wind, and it constantly changes the shape of the landscape. Erosion happens more quickly on bare rock, which is unprotected by soil. It often begins with weathering, where rocks are weakened by the weather’s elements, such as sunshine, frost, and rain. Rocks can be eroded by physical weathering through heat, cold and frost, and CHEMICAL WEATHERING. Erosion may lead to the MASS MOVEMENT of rock and soil. Waves erode the base of cliffs, undermining them and making them collapse. This can create coastal features such as the Twelve Apostles in Victoria, Australia. The stacks (rock towers) are left when headlands are worn away from both sides until they crumble. The broken rocks form shingle and sand beaches. Erosion happens faster when shingle is thrown against the cliffs by the waves. Mountain ranges contain deep valleys that have been carved out by glaciers. A glacier is like a slow-moving river of ice that flows downhill, carried forwards by its huge weight. The rocks dragged along underneath it gouge deep into the ground, creating U-shaped valleys with steep sides and flat bottoms. Sand blown by strong winds has sculpted the slender sandstone pillars of Bryce Canyon, Utah, USA. Their rugged outlines are caused by the softer layers of rock are being eroded more quickly than the harder layers. Wind erosion is common in deserts, where sand is blown about because there are few plants to hold the soil in place and there is no rain to bind the soil particles together. Some rocks are broken down by chemical action, in a process called chemical weathering. The minerals they contain are changed chemically by the effects of sunlight, air, and especially water. The rocks are weakened and wear away more easily. Limestone, for example, is dissolved by rainwater, because the water contains carbon dioxide from the atmosphere, making it slightly acidic. Erosion normally breaks down the landscape a tiny piece at a time, but sometimes rocks and soil move downhill in large volumes. These movements, which include landslides, mudflows, and rock falls, are called mass movements. They happen when rock, debris, or soil on a slope becomes unstable and can no longer resist the downward force of gravity. Soil creep is the extremely slow movement of soil down a steep hillside. It is caused by soil expanding and contracting, when it goes from wet to dry or frozen to unfrozen. The top layers of the soil move faster than the layers underneath. The movement is far too slow to see, but bent trees, leaning fence posts and telegraph poles, and small terraces in fields are all evidence of soil creep. Soil may also build up against a wall or at the bottom of the hillside. A slump is a mass movement that happens when a large section of soil or soft rock breaks away from a slope and slides downwards. Short cliffs called scarps are left at the top of the slope. Slumps often happen where the base of a slope is eroded by a river or by waves, or when soil or soft rock becomes waterlogged. A lahar is a mudflow of water mixed with volcanic ash. This forms when ash mixes with melting ice during an eruption, or with torrential rain. The mud flows down river valleys and sets hard when it comes to a stop. Lahars can cause destruction on a massive scale. Debris is made up of broken rock, sometimes mixed with soil. These pieces of debris may collect on a slope and begin to roll or slide downwards. Debris slides often happen where people have cleared hillsides of trees and other vegetation, which causes the soil and rock to be eroded quickly.

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Removal Act of 1830 On May 26, 1830, the Indian Removal Act of 1830 was passed by the Twenty-First Congress of the United states of America. After four months of strong debate, Andrew Jackson signed the bill into law. Land greed was a big reason for the federal government's position on Indian removal. This desire for Indian lands was also abetted by the Indian hating mentallity that was peculiar to some American frontiersman. This period of forcible removal first started with the Cherokee Indians in the state of Georgia. In 1802, the Georgia legislature signed a compact giving the federal government all of her claims to western lands in exchange for the government's pledge to extigiush all Indian titles to land within the state. But by the mid-1820's Georgians began to doubt that the government would withhold its part of the bargain. The Cherokee Indian tribes had a substantial part of land in Georgia that they had had for many generations though. They were worried about losing their land so they forced the issue by adopting a written constitution. This document proclaimed that the Cherokee nation had complete jurisdiction over its own territory. But by now Indian removal had become entwined with the state of Georgia's rights and the Cherokee tribes had to make their claims in court. When the Cherokee nation sought aid from newly elected president Andrew Jackson, he informed them that he would not interfere with the lawful prerogatives of the state of Georgia. Jackson saw the solution of the problem with the removal of the Cherokee tribes to lands west. This would keep contact between Indians and colonists rare. He suggested that laws be past so that the Indians would have to move west of the Mississippi river. Similar incidents happened between the other "civilized" tribes and white men. The Seminole tribe had land disputes with the state of Florida. The Creek Indians fought many battles against the federal army so they could keep their land in the states of Alabama and Georgia. The Chickisaw and Choctaw had disputes with the state of Mississippi. To ensure peace the government forced these five tribes called the Five Civilized Tribes to move out of their lands that they had lived on for generations and to move to land given to them in parts of Oklahoma. Andrew Jackson was quoted as saying that this was a way of protecting them and allowing them time to adjust to the white culture. This land in Oklahoma was thinly settled and was thought to have little value. Within 10 years of the Indian Removal Act, more than 70,000 Indians had moved across the Mississippi. Many Indians died on this journey. "The Trails of Tears" The term "Trails of Tears" was given to the period of ten years in which over 70,000 Indians had to give up their homes and move to certain areas assigned to tribes in Oklahoma. The tribes were given a right to all of Oklahoma except the Panhandle. The government promised this land to them "as long as grass shall grow and rivers run." Unfortunately, the land that they were given only lasted till about 1906 and then they were forced to move to other reservations. The Trails of Tears were several trails that the Five civilized Tribes traveled on their way to their new lands. Many Indians died because of famine or disease. Sometimes a person would die because of the harsh living conditions. The tribes had to walk all day long and get very little rest. All this was in order to free more land for white settlers. The period of forcible removal started when Andrew Jackson became Presidentin 1829. At that time there was reported to be sightings of gold in the Cherokee territory in Georgia which caused prospectors to rush in, tearing down fences and destroying crops. In Mississippi, the state laws were extended over Choctaw and Chickisaw lands, and in 1930 the Indians were made citizens which made it illegal to hold any tribal office. Also in Georgia, the Cherokee tribes were forbade to hold any type of tribal legislature except to ratify land cessions, and the citzens of Georgia were invited to rob and plunder the tribes in their are by making it illegal for an Indian to bring suit against a When President Jackson began to negotiate with the Indians, he gave them a guarantee of perpetual autonomy in the West as the strongest incentive to emigration. The Five tribes gave all of their Eastern lands to the United States and agreed to migrate beyond the Mississippi by the end of the 1830's. The Federal agents accomplished this by bribery, trickery,and intimidation. All of the treaties signed by the Indians as the agreed to the terms of the removal contained guarantees that the Indians, territory should be perpetual and that no government other than their own should be erected over them without their consent. The land retained by the five civilized tribes was known as the Indian Territory. The 19,525,966 acres were divded among the the five tribes. The Choctaws received 6,953,048 acres in the southeast part of Oklahoma; the Chickisaw recieved over 4,707,903 acres west of the Choctaws reservation; the Cherokees received 4,420,068 acres in the northeast; the received 3,079,095 acres southwest of the Cherokees; and the Seminoles purchased 365,852 acres which they purchased from their kin, the Creeks. The Chickisaw and the Choctaw owned their lands jointly because they were so closely related but the tribes still exercised jurisdiction over its own territory though. Besides the land that the tribes obtained, they also received a large sum of money fom the sale of its Eastern territories. This money was a considerable part of the revenue for the tribes and was used by their legislatures for the support of schools and their governments. The Cherokee nation held $2,716,979.98 in the United States trust; the Choctaw nation had $975,258.91; the Chickisaw held __BODY__,206,695.66;the Creek had $2,275,168.00; and the Seminole had $2,070,000.00 by the end of 1894. After the end of the Trails of Tears, the conversion tof all tribes to Christianity had been efected rapidly. The Seminoles and Creeks were conservative to their customs but other tribes were receptive to any custom considered supperior to their own. The tribes found Christian teachings fitted to their own. Mainly the modernization change began at the end of the removal. Andrew Jackson Gave a speech on the Indian removal in the year of 1830. He said, "It gives me great pleasure to announce to Congress that the benevolent policy of the government, steady pursued for nearly thirty years, in relation with the removal of the indians beyond the white settlements is approaching to a happy "The consequences of a speedy will be important to the United States, to individual states, and to the Indians themselves. It puts an end to all possible danger of a collision betweewn the authorities of the general and state governments, and of the account the Indians. It will place a dense population in large tracts of country now occupied by a few savaged hunters. By opening the whole territory between Tenesee on the north and Louisiana on the south to the settlement of the whites it will incalcuably strengthen the Southwestern frontier and render the adjacent states strong enough to repel future invasion without remote aid." "It will seperate the indians from immediate contact with settlements of whites; enable them to pusue happiness in their own way and under their own rude institutions; will retard the progress of decay, which is lessening their numbers, and perhaps cause them gradually, under the protection of the government and through the influences of good counsels, to cast off their savage habits and become an interesting, civilized, and christian community." For two decades Fort Gibson was the base of operations for the American army as they tried to keep the peace. During the 1810's to 1830's, John C. Calhoun, James Monroe's secretary of war, tried to relocate several Eastern tribes beyond the area of the white settlements. Fort Gibson was brought up because it served as barracks for the army. The relocation area for the Eastern tribes was part of other tribes land. The other tribes wanted toprotect it so they fought for it. The soldiers from Fort Gibson began to make boundaries, construct roads, and escort delegates to the region. The soldiers also started to implement the removal process in other ways to. The soldiers of Fort Gibson were fiercly hated by the Indian tribes of that region. Yet during the many years of the indian removal, there was never a alsh between the soldiers or the tribes. An Indian was never killed by the Army. The soldiers at Fort Gibson served as a cultural buffer between the whites and the indians. The Fort was established in the 1820's by General Matthew Arbuckle. He served and commanded it through most of it's two decades during the Indian removal. He wrote his last report from it on June 21, 1841. THE CHEROKEE INDIANS The Cherokee Indians live in many parts of the United states, but more than 100,000 live in parts of Oklahoma. Many Cherokee have moved elsewhere. In the 1800's, the Cherokee Nation was one of the strongest Indian tribes in the United States. They were part of the Five Civilized Tribes. The Cherokee Nation began to adopt the economic and political stucture of the white settlers in the early 1800's. They owned large plantations and some even kept slaves. The Cherokee Nation was a form of republican government. A Cherokee Indian named Sequoya introduced a system of writing for the Cherokee language in 1821 also. White settlers began to protest the Cherokee's right to own land in the early 1800'. They demanded that the Cherokee Nation be moved west of the Mississippi to make room for white settlers. Some members of the Cherokee Nation signed treaties with the government in 1835 agreeing to move to designated areas in Oklahoma. Most of the tribe did not want to be relocated so they opposed the treaty. But most of the Cherokees, led by Chief John Ross, were forced to move to the Indian Territory in the winter of 1838-1839. More than 17,000 Cherokees marched from their homes to Oklahoma. This march was called the Trail of Tears. Many Indians died on this journey. Even though most of the Cherokee nation been forced to move, more than a 1,000 Cherokee escaped and remained in the Great Smoky Mountains, which is in parts of Tenessee and North Carolina. These tribes became known as the Eastern Band of Cherokee. The Cherokee who went west reformed the political system that they had before. The Cherokee Nation set up schools and churches. But all this progress was stopped in the late 1800's. Congress voted to abolish the Cherokee Nation to open yet more land for settlement by whites. Today most of the Cherokee remain in northeastern Oklahoma, where they have reestablished their form of government. The Chickisaw Indians were a tribe that lived in the southern United States. Their land included western Tenessee and Kentucky, northwestern Alabama, and northern Mississippi before the Indian removal. They were relocated to Oklahoma by the government in the 1830's. The Chickisaws lived in several small vilages with one- room log cabins. The people supported meach other by trading with other tribes, fishing, farming, and hunting. Each village was headed by a chief. The Chickisaw Indians were known as fierce warriors. They fought for Great Britain when they fought France and Spain for control of the southern United States. They also helped them fight against the colonists in the Revolutionary War (1775-1783). And During the Civil War, the tribe fought for the Confederacy (1861-1865). The tribe was relocated to the Indian Territory in 1837 by the National Government. They also took part in the Trail of Tears. In 1907, the Chickisaw Indian territory became part of the new state of Oklahoma. About 5,300 Chickisaw descendants live in Oklahoma. They have a Democratic government in which they elect their leaders for the welfare of the tribe. The Choctaw tribe originates from Alabama and Mississippi. They believed in the primitive ways and hunted and farmed to support themselves. They raised corn and other crops to trade with other Indians. They celebrate their crops with their chief religious ceremony which is a harvest celebration called the Green Corn Dance. One of their legends states that the Choctaw Indian tribe was created at a sacred mount called Nanih Waiya, near Noxapater, Mississippi. After the Indian Removal Act was passed, the Choctaw Indians were forced to move west in order to make room for more white settlers. They were forced to sighn the Treaty of Dancing Rabbit Creek after fierce fighting with the United States army. This treaty exchanched the Indians land for the assigned Indian Territory in what is now Oklahoma. In the early 1830's, over 14,000 Choctaws moved to the Indian Territory in several groups. Although many groups of Indians were gone, over 5,000 Choctaws remained in The Choctaws who moved to the Indian Territory established their own way of life. They modernized themselves by establishing schools and an electoral form of government. In the Civil War, the Choctaw Indians fought on the side of the Confederacy and when the south was defeated, they were forced to give up much of their land. Their tribal governments were dissolved by 1907, when Oklahoma became a state. It stayed that way unttil 1970 when they were recognized by congress and allowed to elect their own chief. Today, many Choctaw are farmers. About 11,000 still live in Oklahoma and nearly 4,000 still live in Mississippi as a seperate tribe. The Creek Indians a part of a 19 tribal group that once resided in much of what is now Alabama and Georgia. Today, many of the 20,000 Creek Indians live in Oklahoma. The Muskogee and the Alabama are the largest Creek tribes. Most of them live north of the other Creek tribes. They are called the Upper Creeks. The lower Creek tribes belong to either Yuchi or Hitichi tribes. In the 1800's, the Creeks fought wars with people trying to settle on their lands. They fought in the first and second Creek Wars. They were great warriors who attacked with the element of surprise. After the Battle of Horseshoe Bend, the Creeks were forced to sign a Treaty that made them give up their land. In the 1830's, they were forced to move to the Indian Territory in what is know Oklahoma. Very few Indians were left behind and they ones who did leave had to leave their belongings behind. The Creeks recieved very little payment for their lands. The Creeks were forced to live in poverty for many years. Many Creeks are still very poor today. Some struggled with crops and became fairly prosperous. Much of the land given to them was not of much value. Also in 1890, a series of laws broke up many tribal landholdings of the Creeks and they were sold to individual Indians. After this, many Creeks were forced back into poverty. The Seminole Indians are a tribe the used to reside in Florida in the early 1800's. The Seminole originally belonged to the Creek tribe. They broke apart from them and moved out of Alabama and Georgia and moved into Florida in the 1700's They became known as Seminoles because the name means runaways. The Seminoles opposed the United States when they came for the Seminole's land. The United acquired Florida in 1819, and began urging them to sell their land to the government and to move to the Indian Territory along with the other southeasten tribes. In 1832, some of the Seminole leaders signed a treaty and promised to relocate. The Seminole tribe split at this time. After the Indians that agreed to move had gone the other part of the tribe fought to keep their lands. They fled into the Florida swamps. They started the Second Seminole war (1835). This was fought over the remaining land that the Seminole had fled to. It lasted for seven years. 1,500 American men died and the cost to the United States was $20 million. The Seminole were led by Osceola until he was tricked by General Thomas Jessup. Osceola was seized and imprisoned by Jessup during peace talks under a flag of truce. Osceola died in 1838 when he still in prison. After the war, many Seminoles moved west but still asmall group stayed hidden in the Florida swamps.

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In the 1950s a mountain range was discovered beneath the ice in Antarctica. The mountains were named the Gamburtsevs. Recently, scientists used ice-penetrating radar and a network of seismometers (size-ma-me-ters) to determine the size and shape of the range. Scientists have learned that the mountains are similar in size and shape to the Alps, with sharp peaks and deep valleys. Previously, secular scientists assumed the Gamburtsevs would be mostly flat. They believed that Antarctica’s ice sheet formed slowly, over millions of years. The steep and jagged mountain peaks would be worn down as the glaciers continually scraped across the mountaintops. Finding such high, jagged peaks under the ice shows that the ice formed quickly. This fits with the creationist model of a catastrophic Ice Age. (An ice age is defined as a time of extensive glacial activity during which a large part of the land is covered by ice.) Most creationists agree that there was one major Ice Age following the Flood. It occurred largely because volcanic ash in the air blocked the sun’s rays, making the summers much cooler. This combined with warm oceans to bring about lots of precipitation, which fell over the high areas and formed ice sheets. Secular scientists have a hard time coming to agreement about ice ages. They have proposed that there were anywhere from four to thirty different times when glaciers covered much of the land. A catastrophic Ice Age would explain why the huge Antarctic mountains were not worn away but were instead covered rapidly. This is another example of how true science confirms what the Bible has already told us!

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Hearing When we hear, we are processing sound waves that are made up of compressions of air or water. We hear vibrations in the air as they strike a part of the ear called the eardrum and make it vibrate. These vibrations are sent through other parts of the ear and finally sent as action potentials to the brain. Sound waves have both amplitude and frequency. Amplitude is a sound’s intensity, and loudness is the perception of that intensity. Frequency of a sound is the number of compressions per second. Pitch is closely related to frequency. Hearing – Outer Ear The part of the hearing system that we see on the outside of the head is called the pinna (the ear). It is designed to capture sound. When a sound reaches the ear, it passes through the tube called the external auditory canal until it reaches the tympanic membrane or eardrum. Hearing – Middle Ear The eardrum vibrates at the same frequency as the sound waves that hit it. Attached to the eardrum are three very small bones (the smallest bones in the body!) that also vibrate to the frequency of the sound. These bones are known as the hammer, anvil, and stirrup because of their shapes. Together, they are known as the ossicles. The three bones are attached to the oval window. The oval window is the beginning of the inner ear. Hearing – Inner Ear The inner ear has the cochlea, a snail-shaped fluid-filled structure. Vibrations from sound in the fluid in the cochlea displace hair cells that are the neuron receptors for sound at the bottom of the cochlea in the basilar membrane. The tectorial membrane covers the hair cells and protects them. The hair cells send signals to the auditory nerve, which sends a signal about sound to the temporal lobe of the brain. Visualization of the Ear Theories about Hearing There are three theories about hearing. The first theory is known as the frequency theory. This theory says that the basilar membrane that holds the hair cells vibrates at the same frequency as sound. This causes the auditory nerve axons to produce action potentials at the same frequency. However, the maximum firing rate of a neuron is short of the highest frequencies we can hear. Theories about Hearing The second theory is known as the place theory. This theory suggests that the basilar membrane is similar to the strings of a piano and that each area along the membrane is tuned to a specific frequency and vibrates to that frequency. The nervous system would have to decide among the frequencies based on which neurons are active. However, the problem with this theory is that some parts of the basilar membrane are bound together too tightly for any part to vibrate like a piano string. Theories about Hearing The final theory is known as the volley principle. This theory suggests we use methods that combine aspects of the frequency theory and the place theory. The basilar membrane is stiff at its base where the stirrup connects with the cochlea and floppy at the other end of the cochlea. Hair cells along the basilar membrane would act differently depending on their location. When we hear sounds at a very high frequency, we use something like the place theory. When we hear lower pitched sounds, we use something like the frequency theory. So combining parts of the first two theories explains how we hear better than using either one of the first two theories separately. Hearing and the Brain Information about hearing, just like information about vision, is first routed through the thalamus and other brain areas below the cortex before reaching the primary auditory cortex, located in the temporal lobe of the cerebral cortex. Different areas of the auditory cortex, just like is true in the visual cortex, process information in different ways, including about the location of a sound and the motion of sound. And just like vision, hearing requires a certain amount of experience with sounds for our hearing to be fully developed. Areas of the brain associated with vision and hearing

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Reading to Learn: Comprehension Strategies Rationale: In order to gain insight while reading one must be able to comprehend. However, students often fail to comprehend (and remember) what they have read. Because of this, teachers have been given comprehension strategies that can be taught to children in order to give them a helping hand. One such strategy is called story-grammar. The following activity will show children how to use the story-grammar strategy to help them comprehend what they are reading. Materials: You will need two copies of ten different conventional stories (The Orphan Kittens by Margaret Wise Brown, Oliver Finds a Home by Justin Korman, Bambi by Felix Salten, Paul Revere by Irwin Shapiro, etc.) that the children will find interesting, Charlotte's Web by E.B. White (with highlighted passages for modeling), a question-answer sheet, and a pencil. Procedure: 1. Explain to the boy's and girl's that today they will be learning a strategy that will show them how to go about comprehending what they read. 2. Have each child come up and choose a book from your selection (There should be two children throughout the room with the same book). Next pass out two question-answer sheets to each child. Tell the children to put the book and one of the question-answer sheets under their desk to be used at a later date. 3. Explain to the children that you want them to listen to you read Charlotte's Web. Tell them that after you read for a few minutes, you will stop, think about the first question on the sheet, and then answer it. Explain that you want them to answer the same question at their desk. You will do this throughout the reading of the book. 4. Now begin reading. After reading a few paragraphs (Read only the highlighted passages for modeling) ask the children to answer the first question: Who are the main characters? Give them two or three minutes to answer, and then read on. After you have read a few more passages, ask them to answer the second question: Where and when did the story take place? Continue this until you have read the entire story and the children have answered the other three questions consisting of What did the main characters do?, How did the story end?, and How did the main character feel? Be sure to let the children know there is no right or wrong answer for the question How did the main character feel? because itís is an open-ended question (Depends on readers interpretation of the story). Now have a class discussion about their answers. 5. The second part of the lesson requires the children to read silently (which is very good for a lot of reading skills such as fluency and comprehension) at their desk for ten minutes. (Model reading silent by telling the students to read by thinking the words in their head without saying them out loud). Ask the children to take out the book that they choose earlier along with the second question-answer sheet. Explain to them how you will set a timer for ten minutes, during which time they are to be reading the book they choose. When the ten minutes are up, ask the children to answer the first question on the sheet in front of them. Give them ample time to reflect on what they have read and then set the timer again. Do this throughout the entire book. (You may want to choose shorter stories or treat this as a daily but weekly, meaning they work on the same books all week, assignment). 7. When the children have finished the book and answered all the questions, have them pair up with the other person in the room who read the same book. Explain how you want them to discuss their answers with each other. If there is a disagreement among them, tell them to talk it over to see why. This will help them see how someone else came to their conclusions. Don't forget to read silently along with the children. (You may want to have one of your students read the same book that you are reading and then discuss it with them. If you do this, write your answers in their language). (This will keep them from feeling over-powered). 8. For a review you can ask the children what five questions they should ask themselves, while reading, to help them comprehend what they have read. References: Pressley, M., Johnson, C. J., Symons, S., McGoldrick, J. A., & Kurity, J. A. (1989). "Strategies That Improve Childrenís Memory and Comprehension of Text. The Elementary School Journal, (1990, Pp. 3-32). Click here to return to Elucidations

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Conjunction Activities for Children This game is great for teaching proper usage of conjunctions. Simply click on the above highlighted link and it will take you to an online crossword puzzle that your students can use. Or, if you would like to make this an in-the-class activity, simply print it off and make copies for everyone. This could be used as a introductory or culminating activity for a conjunction unit. Use this activity as a timed exercise in which you provide the students with fifty simple sentences. The students must then link the sentences together to further comprehend the purpose of conjunctions. In this particular activity students will create compound sentences with the use of conjunctions. The purpose of this assignment is to show the importance of conjunctions and the role that they play in everyday grammar and written form. The objective will be to get students to write two paragraphs using a variety of simple and compound sentences. This is a great way to get your students to visualize the function that conjunctions perform! Try it out, and let us know if it helped! This activity will allow students to create their own sentences using the conjunction "But". Students will divide into teams and then create sentences starting with "Yes, but...."

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The first spacecraft to globally map the Moon left lunar orbit on May 3, 1994. Clementine, a joint Department of Defense-NASA mission, had systematically mapped the Moon’s surface over 71 days, collecting almost 2 million images. For the first time, scientists could put results of the Apollo lunar sample studies into a regional, and ultimately, a global context. Clementine collected special data products, including broadband thermal, high resolution and star tracker images for a variety of special studies. But in addition to this new knowledge of lunar processes and history, the mission led a wave of renewed interest in the processes and history of the Moon, which in turn, spurred a commitment to return there with both machines and people. We peeked into the Moon’s cold, dark areas near the poles and stood on the edge of a revolution in lunar science. Prior to Clementine, good topographic maps only existed for areas under the ground tracks of the orbital Apollo spacecraft. From Clementine’s laser ranging data, we obtained our first global topographic map of the Moon. It revealed the vast extent and superb preservation state of the South Pole-Aitken (SPA) basin and confirmed many large-scale features mapped or inferred from only a few clues provided by isolated landforms. Correlated with gravity information derived from radio tracking, we produced a map of crustal thickness, thereby showing that the crust thins under the floors of the largest impact basins. Two cameras (with eleven filters) covered the spectral range of 415 to 1900 nm, where absorption bands of the major lunar rock-forming minerals (plagioclase, pyroxene and olivine) are found. Varying proportions of these minerals make up the suite of lunar rocks. Global color maps made from these spectral images show the distribution of rock types on the Moon. The uppermost lunar crust is a mixed zone, where composition varies widely with location. Below this zone is a layer of nearly pure anorthosite, a rock type made up solely of plagioclase feldspar (formed during the global melting event that created the crust). Craters and large basins act as natural “drill holes” in the crust, exposing deeper levels of the Moon. The deepest parts of the interior (and possibly the upper mantle) are exposed at the surface within the floor of the enormous SPA basin on the far side of the Moon. Clementine showed us the nature and extent of the poles of the Moon, including peaks of near permanent sun-illumination and crater interiors in permanent darkness. From his first look at the poles, Gene Shoemaker (Leader of the Clementine Science Team) got an inkling that something interesting was going on there. Gene was convinced that water ice might be present, an idea about which I had always been skeptical. At that time, no trace of hydration had ever been found in lunar minerals and the prevailing wisdom was that the Moon is now and always had been bone dry. With Gene arguing to keep an open mind and Stu Nozette (Deputy Program Manager) devising a bistatic radio frequency (RF) experiment to use the spacecraft transmitter to “peek” into the dark areas of the poles, we moved ahead on planning the observations. To my astonishment (and delight), a pass over the south pole of the Moon showed evidence for enhanced circular polarization ratio (CPR) – a possible indicator of the presence of ice. A control orbit over a nearby sunlit area showed no such evidence. However, CPR is not a unique determinant for ice, as rocky, rough surfaces and ice deposits both show high CPR. It took a couple of years to reduce and fully understand the data, but the bistatic experiment was successful. In part, our ice interpretation was supported by the discovery of water ice at the poles of Mercury (a planet very similar to the Moon). We published our results in Science magazine in December 1996, setting off a media frenzy and a decade of scientific argument and counter-argument about the interpretation of radar data for the lunar poles (an argument that continues to this day, despite subsequent confirmation of lunar polar water from several other techniques). Along with Clementine’s success came a growing interest in lunar resources and a new appreciation for the complexity of the Moon. This interest led to the selection of Lunar Prospector (LP) as the first PI-led mission of NASA’s new, low-cost Discovery series of planetary probes. LP flew to the Moon in 1998 and carried instruments complementary to the data produced by Clementine, including a gamma-ray spectrometer to map global elemental composition, magnetic and gravity measurements, and a neutron spectrometer to map the distribution of hydrogen. LP found enhanced concentrations of hydrogen at both poles, again suggesting that water ice was probably present. The debate on the abundance and physical nature of the water ice continued, with estimates ranging from a simple enrichment of solar wind implanted hydrogen in polar soils, to substantial quantities of water ice trapped in the dark, cold regions of the poles. Buttressed by this new information, the Moon became an attractive destination for robotic and human missions. With direct evidence for significant amounts of hydrogen (regardless of form) on the surface, there now was a known resource that would support long-term human presence. This hydrogen discovery was complemented by the identification in Clementine images of several areas near the pole that remain sunlit for substantial fractions of the year – not quite the “peaks of eternal light” first proposed by French astronomer Camille Flammarion in 1879 but something very close to it. The availability of material and energy resources – the two biggest necessities for permanent human presence on the Moon – was confirmed in one fell swoop. Combined, the results of Clementine and LP finally gave scientists the Lunar Polar Orbiter mission we had long sought. These two missions certified the possibility of using lunar resources to provision ourselves in space, permanently establishing the Moon as a valuable, enabling asset for human spaceflight. Remaining was to verify and extend the radar results from Clementine and map the ice deposits of the poles. The Clementine bistatic experiment led to the development of an RF transponder called Mini-SGLS (Space Ground Link System), which flew on the Air Force mission MightySat II in 2000. This experiment miniaturized the RF systems necessary for a low mass, low power imaging radar. With the 2008 inclusion of our Mini-SAR on India’s Chandryaan-1 lunar orbiter, we finally got the chance to build and fly such a system. Chandrayaan-1 not only mapped the high CPR material at both poles, it also carried a spectrometer (the Moon Mineralogy Mapper, or M3) that discovered large amounts of adsorbed surface water (H2O) and hydroxyl (OH) at high latitudes. Coupled with the measurement of exospheric water above the south pole by its Moon Impact Probe, Chandrayaan-1 significantly advanced our understanding of polar water, revealing it to be abundant and present in more varied forms on the Moon than had previously been imagined. The ever increasing weight of evidence for the presence of significant amounts of water at the lunar poles led to the LCROSS experiment being “piggybacked” on NASA’s 2008 Lunar Reconnaissance Orbiter (LRO) mission. LCROSS was a relatively inexpensive add-on, designed to observe the collision of the LRO launch vehicle’s Centaur upper stage with the lunar surface, looking for water in the ejecta plume of that impact. Water in both vapor and solid form was observed, suggesting the presence of water ice in the floor of the crater Cabaeus (at concentration levels between 5 and 10 weight percent). LRO orbits the Moon and collects data to this day. Although much remains unknown about lunar polar water, we now know for certain that it exists; such knowledge has completely revised our thinking about the future use and habitation of the Moon. The Clementine programmatic template has influenced spaceflight for the last 20 years. The Europeans flew SMART-1 to the Moon in 2002, largely as a technology demonstration mission with goals very similar to those of Clementine. NASA directed the Applied Physics Laboratory (APL) to fly Near-Earth Asteroid Rendezvous (NEAR) to the asteroid Eros in 1995 as a Discovery mission, attaining the asteroid exploration opportunity missed when control of the Clementine spacecraft was lost after leaving the Moon. India’s Chandrayaan-1 was of a size and payload scope similar to Clementine. The selection of LCROSS as a low-cost, fast-tracked, limited objectives mission further extended use of the Clementine paradigm. The “Faster-Better-Cheaper” mission model, once panned by some in the spaceflight community, is now recognized as a preferred mode of operations, absent the emotional baggage of that name. A limited objectives mission that flies is more desirable than a gold-plated one that sits forever on the drawing board. While some missions do require significant levels of fiscal and technical resources to attain their objectives, an important lesson of Clementine is that for most scientific and exploration goals, “better” is the enemy of “good enough.” Space missions require smart, lean management; they should not be charge codes for feeding the beast of organizational overhead. Clementine was lean and fast; perhaps we would have made fewer mistakes had the pace been a bit slower, but overall the mission gave us a vast, high-quality dataset, still extensively used to this day. The Naval Research Laboratory transferred the Clementine engineering model to the Smithsonian in 2002. The spacecraft hangs today in the Air and Space Museum, just above the Apollo Lunar Module. It is probably not too much of an exaggeration to say that Clementine changed the direction of the American space program. After the failure of SEI in 1990-1992, NASA was left with no long-term strategic direction. For the first time in its history, NASA had no follow-on program to Shuttle-Station, despite attempts by Dan Goldin and others to secure approval for a human mission to Mars (then and now, a bridge too far – both technically and financially). This programmatic stasis continued until 2003, when the tragic loss of Columbia led to a top-down review of U.S. space goals. Because Clementine had documented its strategic value, the Moon once again became an attractive destination for future robotic and human missions. The resulting Vision for Space Exploration (VSE) in 2004 made the Moon the centerpiece of a new American effort beyond low Earth orbit. While Mars was vaguely discussed as an eventual (not ultimate) objective, the activities to be done on the Moon were specified in detail in the VSE, particularly with regard to the use of its material and energy resources to build a sustainable program. Regrettably, various factors combined to subvert the Vision, thereby ending the strategic direction of America’s civil space program. Clementine was a watershed, the hinge point that forever changed the nature of space policy debates. A fundamentally different way forward is now possible in space – one of extensibility, sustainability and permanence. Once an outlandish idea from science fiction, we have found that lunar resources can be used to create new capabilities in space, a welcome genie that cannot be put back in the bottle. Americans need to ask why their national space program was diverted from such a sustainable path. We cannot afford to remain behind while others plan and fly missions to understand and exploit the Moon’s resources. Our path forward into the universe is clear. In order to remain a world leader in space utilization and development – and a participant in and beneficiary of a new cislunar economy – the United States must again direct her sights and energies toward the Moon. Note: Background history for the Clementine mission is described in a companion post at my Spudis Lunar Resources blog.

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The first African-American Member of Congress was elected nearly 100 years after the United States became a nation. Slavery had only been illegal for five years in the American South when Representative Joseph Rainey of South Carolina and Senator Hiram Revels of Mississippi were elected to office in 1870. In fact, the states they served had been represented by slave owners only 10 years earlier. The early African-American Members argued passionately for legislation promoting racial equality, but it would still be many years before they would be viewed as equals. On December 5, 1887, for the first time in almost two decades, Congress convened without an African-American Member. For nearly 30 years, no African Americans served in Congress. With his election to the U.S. House of Representatives from a Chicago district in 1928, Oscar De Priest of Illinois became the first African American to serve in Congress since George White of North Carolina left office in 1901. Use the interactive map to compile information on the representation of Black Americans in Congress, such as the number of Members who served from a particular state or region and when they served.

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As discussed in the previous part of Lesson 3, the slope of a position vs. time graph reveals pertinent information about an object's velocity. For example, a small slope means a small velocity; a negative slope means a negative velocity; a constant slope (straight line) means a constant velocity; a changing slope (curved line) means a changing velocity. Thus the shape of the line on the graph (straight, curving, steeply sloped, mildly sloped, etc.) is descriptive of the object's motion. In this part of the lesson, we will examine how the actual slope value of any straight line on a graph is the velocity of the object. Consider a car moving with a constant velocity of +10 m/s for 5 seconds. The diagram below depicts such a motion. The position-time graph would look like the graph at the right. Note that during the first 5 seconds, the line on the graph slopes up 10 m for every 1 second along the horizontal (time) axis. That is, the slope of the line is +10 meter/1 second. In this case, the slope of the line (10 m/s) is obviously equal to the velocity of the car. We will examine a few other graphs to see if this a principle that is true of all position vs. time graphs. Now consider a car moving at a constant velocity of +5 m/s for 5 seconds, abruptly stopping, and then remaining at rest (v = 0 m/s) for 5 seconds. If the position-time data for such a car were graphed, then the resulting graph would look like the graph at the right. For the first five seconds the line on the graph slopes up 5 meters for every 1 second along the horizontal (time) axis. That is, the line on the position vs. time graph has a slope of +5 meters/1 second for the first five seconds. Thus, the slope of the line on the graph equals the velocity of the car. During the last 5 seconds (5 to 10 seconds), the line slopes up 0 meters. That is, the slope of the line is 0 m/s - the same as the velocity during this time interval. Both of these examples reveal an important principle. The principle is that the slope of the line on a position-time graph is equal to the velocity of the object. If the object is moving with a velocity of +4 m/s, then the slope of the line will be +4 m/s. If the object is moving with a velocity of -8 m/s, then the slope of the line will be -8 m/s. If the object has a velocity of 0 m/s, then the slope of the line will be 0 m/s. The widget below plots the position-time plot for an object moving with a constant velocity. Simply enter the velocity value, the intial position, and the time over which the motion occurs. The widget then plots the line with position on the vertical axis and time on the horizontal axis.

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In the last chapter, we graphed data. Now, we move to graphing equations with two variables. For simplicity, the discussion in this chapter is confined to linear equations, i.e. equations of degree 1 . Some of the general concepts carry over to more general equations, to be discussed later. The first section explains how to represent variables as ordered pairs. This is a convenient way of writing corresponding variable values. In this section, we will also learn how to graph ordered pair values (x, y) on an xy-graph. Graphing (x, y) values on a graph is similar to graphing x values on a number line, except that we are working in two dimensions instead of one. The second section provides an introduction to graphing equations. It explains how to make a data table of (x, y) values and how to make a graph from a data table. There are several methods of graphing equations. The next section introduces another method of graphing linear equations using the x-intercept and y-intercept. It is similar to creating a data table, but often quicker. The fourth section explains the concept of slope. Slope is a characteristic of a linear equation that will allow us to graph that linear equation, recognize its graph, and understand how it relates to other linear equations. The final section introduces a third method of graphing linear equations, which uses slope. It explains how to graph a linear equation given its slope and a single point, and it explains how to determine the slope of a line, given its equation. Graphing is an enormous topic in algebra I and algebra II. No matter what type of equations you study in future algebra, you will probably need to know how to graph them. Thus, it is important to understand the material in this introductory chapter. Each method of graphing learned here will become useful in later topics in algebra, pre- calculus, and even calculus. Graphing also has practical applications. Chemists and physicists use graphs to discover relationships between quantities. Graphs can be used to predict future values of important figures like population and the national debt. Graphs are used in almost every discipline, so it is important to develop an understanding of how to use them.

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The space near black holes is one of the most extreme environments in the Universe. The bodies' strong gravity and rotation combine to create rapidly spinning disks of matter that can emit huge amounts of light at very high energies. However, the exact mechanism by which this light is produced is uncertain, largely because high-resolution observations of black holes are hard to do. Despite their outsized influence, black holes are physically small: even a black hole a billion times the mass of the Sun occupies less volume than the Solar System. A new X-ray observation of the region surrounding the supermassive black hole in the Great Barred Spiral Galaxy may have answered one of the big questions. G. Risaliti and colleagues found the distinct signature of X-rays reflecting off gas orbiting the black hole at nearly the speed of light. The detailed information the astronomers gleaned allowed them to rule out some explanations for the bright X-ray emission, bringing us closer to an understanding of the extreme environment near these gravitational engines. Despite the stereotype of black holes "sucking" matter in, they attract it via gravity. That means stars, gas, and other things can fall into orbits around black holes, which may be stable for long periods of time. Gas often forms accretion disks and jets that release huge amounts of energy in the form of light. This energy can include X-ray emissions. So despite their name, black holes can be very luminous objects. Nearer the boundary of a rotating black hole—its event horizon—the strength of gravity is such that the space matter occupies can be also dragged around the black hole. This effect is called "frame dragging," and is predicted by Einstein's general theory of relativity. The region in which frame dragging becomes significant, however, is very close to the black hole's event horizon, which is relatively small, especially when imaged from Earth. As a result, astronomers could not be sure whether ordinary orbital effects or relativistic frame-dragging is more important for producing the intense X-ray emissions. Astronomers paid particularly close attention to the supermassive black hole at the center of the Great Barred Spiral Galaxy (also known by its catalog number NGC 1365) when a cloud of gas momentarily eclipsed it. That rare event allowed them to get a good size estimate for the accretion disk that surrounds the black hole. The current study followed up by monitoring fluctuations in the X-ray emissions, using the orbiting XMM-Newton and NuSTAR X-ray telescopes. In particular, the researchers looked at emission from neutral and partly ionized iron atoms in the gas. Prior observations showed that the emission lines were broadened, which can be caused by several different phenomena. Researchers considered two primary hypotheses: absorption by other gas along the line of sight between the black hole and us, or very fast motion of the gas itself. The new data strongly supported the latter option. In this scheme, the observed X-ray light reflected off the inner edge of the accretion disk, where the gas is moving at very close to the speed of light. According to the models, this scattering occured well within the frame-dragging region near the black hole. The inner edge of the accretion disk may be close to or at the minimum stable distance from the black hole. Closer than that distance, and matter can no longer orbit in a circular path—it will tend to spiral in. The authors argued that any explanation of the X-ray emission that fails to account for the general-relativistic effects just won't work. Previous observations estimated that the black hole in the Great Barred Spiral Galaxy is spinning nearly as fast as possible; whether other black holes will have similar properties remains an open question.

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These are not simply facts to be memorized. These are complex concepts that students need to develop through engagement with the natural world, through drawing on their previous experiences and existing knowledge, and through the use of models and representations as thinking tools. Students should practice using these ideas in cycles of building and testing models in a wide range of specific situations. At this grade band, students can begin to ask the questions: What is the nature of matter and the properties of matter on a very small scale? Is there some fundamental set of materials from which other materials are composed? How can the macroscopically observable properties of objects and materials be explained in terms of these assumptions? In addition, armed with new insight provided by their knowledge of the existence of atoms and molecules, they can conceptually distinguish between elements (substances composed of just one kind of atom) and compounds (substances composed of clusters of different atoms bonded together in molecules). They can also begin to imagine more possibilities that need to be considered in tracking the identity of materials over time, including the possibility of chemical change. Students have to be able to grasp the concept that if matter were repeatedly divided in half until it was too small to see, some matter would still exist—it wouldn’t cease to exist simply because it was no longer visible. Research has shown that as students move from thinking about matter in terms of commonsense perceptual properties (something one can see, feel, or touch) to defining it as something that takes up space and has weight, they are increasingly comfortable making these kinds of assumptions. This is one example of the ways in which the framework that students developed in the earlier primary and elementary grades prepares them for more advanced theorizing at the middle school level. Middle school science students must conjecture about and represent what matter is like at a level that they can't see, make inferences about what follows from different assumptions, and evaluate the conjecture based on how well it fits with a pattern of results. Research has shown that middle school students are able to discuss these issues with enthusiasm, especially when different models for puzzling phenomena are implemented on a computer and they must judge which models embody the facts. This approach led students who had relevant macroscopic understanding of matter to see the discretely spaced particle model as a better explanation than alternatives (e.g., continuous models and tightly packed particle models). Class discussions allowed students to establish more explicit rules for evaluating

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In late 1863, President Abraham Lincoln and the Congress began to consider the question of how the Union would be reunited if the North won the Civil War. In December President Lincoln proposed a reconstruction program that would allow Confederate states to establish new state governments after 10 percent of their male population took loyalty oaths and the states recognized the “permanent freedom of slaves.” Several congressional Republicans thought Lincoln’s 10 Percent Plan was too mild. A more stringent plan was proposed by Senator Benjamin F. Wade and Representative Henry Winter Davis in February 1864. The Wade-Davis Bill required that 50 percent of a state’s white males take a loyalty oath to be readmitted to the Union. In addition, states were required to give blacks the right to vote. Congress passed the Wade-Davis Bill, but President Lincoln chose not to sign it, killing the bill with a pocket veto. Lincoln continued to advocate tolerance and speed in plans for the reconstruction of the Union in opposition to the Congress. After Lincoln’s assassination in April 1865, however, the Congress had the upper hand in shaping Federal policy toward the defeated South and imposed the harsher reconstruction requirements first advocated in the Wade-Davis Bill.

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The Seasonal Merry-Go-RoundThe tilt of Earth's rotational axis and the Earth's orbit around the Sun work together to create the seasons. As the Earth travels around the Sun, it remains tipped over in the same direction, with its north pole pointed towards the star Polaris. This 23.5 degree tilt of the Earth's rotational axis to the Earth's orbital plane about the Sun (the ecliptic plane) causes Earth's two hemispheres to be exposed to different intensities of sunlight for different amounts of time throughout the year. The changing intensity and changing amount of sunlight to the different hemispheres has given rise to the seasons of summer, fall, winter and spring. In the Northern Hemisphere, the first day of summer, called the summer solstice, is around June 21st. The summer solstice marks the point at which the north pole of the Earth is tilted at its maximum towards the Sun. For any location in the northern hemisphere, the day of the summer solstice is the longest day of the year with the Sun reaching its greatest angular distance north (or its highest point in the sky for the year for that given location). During the time surrounding the summer solstice, the northern hemisphere is getting more direct sunlight for a longer amount of time, which heats that hemisphere efficiently causing warmer temperatures on average. Notice that when the northern hemisphere is tilted towards the Sun, the southern hemisphere is tilted away. This is why people in the North America, Europe, Asia, and other places north of the equator have the opposite season of people in South America, Australia, and other places south of the equator. So, while the summer solstice marks the beginning of summer for the northern hemisphere, it marks the beginning of winter for the southern hemisphere. The first day of winter for the northern hemisphere is called the winter solstice. This day, around December 21st each year, is when the north pole of the Earth is tilted at its maximum away from the Sun. The Sun’s rays are less intense at this time of year because they are spread over a greater surface area and must travel through more energy-absorbing atmosphere to reach the Earth. Also, the winter solstice is the shortest day of the year for those who live in the northern hemisphere. The decreased intensity of the sunlight received along with less daylight hours leads to the cooler temperatures often felt by those living in the northern hemisphere during winter months. Again, things are reversed for the southern hemisphere, where summer is being ushered in at the time of the winter solstice. Of course, between the season of summer and winter in the northern hemisphere comes the season of fall. The beginning of fall in the northern hemisphere is marked by the autumnal equinox (around September 23rd each year). And between the seasons of winter and summer in the northern hemisphere is spring. The beginning of spring in the northern hemisphere is marked by the vernal equinox (around March 21st each year). At the two equinoxes, neither the north pole nor the south pole is inclined toward the Sun. Equinox literally means "equal night". On the vernal (spring) and autumnal (fall) equinoxes, day and night are about the same length all over the world.

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If students are not familiar with or have not recently practiced plotting points on the first quadrant using ordered pairs, review that concept before continuing. Distribute the Coordinate Geometry sheet (M-5-3-3_Coordinate Geometry and KEY.docx) to each student. On a copy of the First Quadrant worksheet from Lesson 2 (M-5-3-2_First Quadrant and KEY.docx), have students plot the points in Figure 1 in order, connecting each point to the previous point and connecting the last point to the first to make a complete, closed shape. Ask students to come up with any words they can think of to describe Figure 1. They may only come up with “triangle.” “There are a lot of names we can call this figure. Shapes have many descriptions, just as you might. You are a person, a boy or girl, maybe a brother or sister, a student, maybe a baseball player, right- or left-handed, and so on. Just as there are many ways to describe you, there are many ways to describe shapes. So, this shape is a triangle since it has three sides, but we can also call it a polygon.” Depending on the class, you can break the word polygon down into two parts: poly- and -gon, and examine each part of the word, explaining that poly- means “many” and -gon means “angles,” and so the word polygon literally means “many angles.” This approach is useful when dealing with other terms like hexagon or octagon (and continues to be useful in higher mathematics when dealing with terms like polynomial). After explaining that a polygon is a figure that has many sides, tell students the sides must be straight lines and the figure must be closed. In other words, they have to connect the last point they plotted back to the first point with a straight line. “Next to the triangle you graphed, write the words polygon and triangle, and then graph Figure 2 on the same coordinate plane on which you graphed Figure 1.” After students have plotted Figure 2, ask them to describe it. Students may respond with rectangle and polygon (or incorrect answers). “What makes this shape a polygon?” (It has many sides, the sides are straight, and the figure is closed.) “What makes this shape a rectangle?” Here, students should focus on the four right angles in the figure. “How many sides does our rectangle have?” (Four) “Just like we have a general name for shapes with three sides—triangle—we also have a general name for shapes with four sides. We call them quadrilaterals.” Possibly write “quadrilaterals” on the board so students can see the term. Again, depending on the class, breaking down the word quadrilateral into parts might be helpful: quad- means “four” and -lateral means “sides.” If students are familiar with football, they may have heard of a lateral pass, which is a pass that goes sideways (as opposed to backward or forward). “So far, then, our shape has a few names. It is a polygon, it is a quadrilateral, and it is a rectangle. It actually has at least one more name. Look at the two long sides that go straight up and down. What word do we have for line segments that will never cross one another no matter how long they are?” (Parallel) “And what about the two short sides on the top and bottom of our rectangle?” (They are also parallel.) “Because our quadrilateral has two pairs of parallel sides, we call it a parallelogram.” Again, write this word on the board so students can see it written out, pointing out the word parallel inside the word parallelogram. Have students write all the terms associated with a parallelogram next to the rectangle. Have students graph Figure 3 on the same coordinate plane as Figures 1 and 2. Ask them to describe it. They should note that it’s a square, a polygon, a quadrilateral, and a parallelogram. If not, ask them if any of the previous terms that applied to rectangle also apply to it. Ask students to explain why the figure is a polygon, quadrilateral, and parallelogram. Lastly, ask them to explain how they know it’s a square. Here, students should focus on both the four right angles and the four sides of equal length. “Now, you said it’s a square because it has four sides of equal length and four right angles. Since it has four right angles, can we also call it a rectangle?” (Yes) “If I ask you to draw a square, can you ever draw one that doesn’t have four right angles?” (No) “So we know that every square is a rectangle.” Have students write down all the terms that apply to the square. Give each student a copy of Quadrilateral Venn diagram sheet (M-5-3-3_Quadrilateral Venn Diagram.docx). Describe how to interpret the diagram (i.e., all squares are rectangles, all rectangles are parallelograms, and all parallelograms are quadrilaterals). Make sure to emphasize that even though all squares are rectangles, for example, there are definitely rectangles (like the one they plotted) that are not squares. On the diagram, illustrate this by identifying the region that is inside the rectangle part of the figure but is outside the square part of the figure. “Write the words “Figure 2” and “Figure 3” on your diagram to show in which part of the diagram they belong.” (Figure 2 belongs in the rectangle portion but not the square portion, while Figure 3 belongs in the square portion.) “Where does Figure 1 go on the diagram?” (Students may respond with Outside the quadrilaterals or not on the diagram.) “We might need another diagram if we want to be able to organize all our polygons. This diagram just organizes quadrilaterals, which have how many sides?” (Four) Before plotting Figure 4, ask students what shape they think it’s going to be. If they aren’t sure (they may be trying to visualize it in their heads), ask them how many points they have to plot. They may at least guess it will have five sides even if they aren’t sure what the figure is called. After discussion, have students plot Figure 4 on the second coordinate plane. “Do any of the words we talked about with the figures on the first coordinate plane apply to this figure?” (Polygon) “We call a five-sided polygon a pentagon.” Again, explaining the meaning of the prefix penta- (five) may be helpful to students. They may also be familiar with the Pentagon in Washington, D.C. (This image: http://www.sciencephoto.com/image/357691/350wm/T8350265-Pentagon_building-SPL.jpg shows the Pentagon from overhead so students can clearly see that it has five sides.) Have students label their pentagon appropriately. Also, explain that whether a polygon is a pentagon is determined only by the number of sides it has. Even though the pentagon in Figure 5 isn’t exactly the same as the Pentagon, they both have five sides and so are both classified as pentagons. “How many sides will Figure 5 have, based on the number of points that need to be graphed?” (Six) Have students plot Figure 5. “What is our six-sided figure called?” Write the word hexagon on the board and explain that hex- means six, so the word literally means “six angles.” Have students label Figure 5 appropriately. Finally, have students plot Figure 6. “How many sides does it have?” (Eight) “What do we call an eight-sided polygon?” If students don’t know, guide them toward the realization that an octopus has eight arms and the prefix oct- means eight, and have them guess what we might call a polygon with eight angles. Have students label the octagon on their coordinate plane appropriately. The Coordinate Plane worksheet can be collected at the end of class and checked against the key to ensure understanding. (Students may use it for reference in Activity 3.) Have students work in pairs for Activity 3. Each student should draw a pattern or design on a coordinate plane that incorporates at least two different polygons, at least one of which should be a quadrilateral. Students should plot each part of their design and include coordinate instructions to provide to their partner. They should label each “set” of coordinates with the name or names of the appropriate polygon. (If they are drawing, for example, a square, they should label the set of coordinates describing the square with the terms square, rectangle, parallelogram, quadrilateral, and polygon.) Once students have listed the coordinates and double-checked their work, they should give their instructions to someone else. “Now, you have the instructions to make someone else’s design. Go ahead and start with the first point on the list and graph each set of points in order. Make sure that the shape you graph matches the name or names the instructions have listed. If you graph something and it’s not a square but the instructions say it is, for example, work with your partner to figure out if you made a mistake in graphing it, your partner made a mistake in writing down the coordinates, or if you both graphed it correctly and it just has the wrong description.” Once students are finished, they should compare their drawings and identify the source of any errors and correct them. This Activity can be repeated if students struggle with writing accurate instructions. Depending on time, to engage students further, they can color and decorate their designs. Use the following strategies to tailor the lesson to meet the needs of your students throughout the year. - Routine: As students explore other geometry topics throughout the year, they can graph the shapes on the coordinate plane, including regular polygons, rhombuses, and even circles (with a designated point as the center and a given radius). They can also explore polygons with more than eight sides, describing them through the use of coordinates. - Small Group: Using larger coordinate planes, students can work in groups to create elaborate designs, with each student responsible for creating the instructions (i.e., listing the coordinates) for part of the design. This activity can be done on large rolls of butcher paper (the coordinate plane can be drawn with a meterstick or yardstick) to create large murals. - Expansion: When working with parallelograms, students can be encouraged to make shapes with parallel sides that are not horizontal or vertical lines. They can explore the idea of slope in the context of “from this point I went to the right 5 units and up 2 units, so from this other point I have to do the same steps,” etc. Students can also be introduced to the distance formula and/or Pythagorean theorem when working on the coordinate plane. Students can also explore the idea of convex and concave polygons through graphing.

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Conditional statements make appearances everywhere. In mathematics or elsewhere, it doesn’t take long to run into something of the form “If P then Q.” Conditional statements are indeed important. What are also important are statements that are related to this conditional statement by changing the position of P, Q and the negation of a statement. Starting with an original statement, we end up with three new conditional statements that are named the converse, the contrapositive and the inverse. Before we define the converse, contrapositive and inverse of a conditional statement, we need to examine the topic of negation. Every statement in logic is either true or false. The negation of a statement simply involves the insertion of the word “not” at the proper part of statement. The addition of the word “not” is done so that it changes the truth status of the statement. It will help to look at an example. The statement “The right triangle is equilateral” has negation “The right triangle is not equilateral.” The negation of “10 is an even number” is the statement “10 is not an even number.” Of course, for this last example we could use the definition of an odd number and instead say that “10 is an odd number.” We note that the truth of a statement is the opposite of that of the negation. We will examine this idea in a more abstract setting. When the statement P is true, the statement “not P” is false. Similarly if P is true, its negation “not P” is true. Negations are commonly denoted with a tilde ~. So instead of writing “not P” we can write ~P. Converse, Contrapositive and Inverse Now we can define the converse, the contrapositive and the inverse for a conditional statement. We start with the conditional statement “If P then Q.” - The converse of the conditional statement is “If Q then P.” - The contrapositive of the conditional statement is “If not Q then notP.” - The inverse of the conditional statement is “If not P then notQ.” We will see how these statements work with an example. Suppose we start with the conditional statement “If it rained last night, then the sidewalk is wet.” - The converse of the conditional statement is “If the sidewalk is wet, then it rained last night.” - The contrapositive of the conditional statement is “If the sidewalk is not wet, then it did not rain last night.” - The inverse of the conditional statement is “If it did not rain last night, then the sidewalk is not wet.” We may wonder why it is important to form these other conditional statements from our initial one. A careful look at the above example reveals something. Suppose that the original statement “If it rained last night, then the sidewalk is wet” is true. Which of the other statements have to be true as well? - The converse “If the sidewalk is wet, then it rained last night” is not necessarily true. The sidewalk could be wet for other reasons. - The inverse “If it did not rain last night, then the sidewalk is not wet” is not necessarily true. Again, just because it did not rain does not mean that the sidewalk is not wet. - The contrapositive “If the sidewalk is not wet, then it did not rain last night” is a true statement. What we see from this example (and what can be proved mathematically) is that a conditional statement has the same truth value as its contrapositive. We say that these two statements are logically equivalent. We also see that a conditional statement is not logically equivalent to its converse and inverse. Since a conditional statement and its contrapositive are logically equivalent, we can use this to our advantage when we are proving mathematical theorems. Rather than prove the truth of a conditional statement directly, we can instead use the indirect proof strategy of proving the truth of that statement’s contrapositive. Contrapositive proofs work because if the contrapositive is true, due to logical equivalence, the original conditional statement is also true. It turns out that even though the converse and inverse are not logically equivalent to the original conditional statement, they are logically equivalent to one another. There is an easy explanation for this. We start with the conditional statement “If Q then P”. The contrapositive of this statement is “If not P then notQ.” Since the inverse is the contrapositive of the converse, the converse and inverse are logically equivalent.

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This is a free lesson from our course in Algebra I In this lesson you learn how to solve linearinequalities and also graph them. An inequality is an algebraic expression with one of these signs: <, >, <=, >=. For example 2x + 3y >= 5. A solution of an inequality is a number which when substituted for the variable makes the inequality a true statement. To graph solution set of linear inequality,for instance,they'd ask you to graph something like x > 2. How did you do it? You would draw your number line, find the "equals" part (in this case, x = 2), mark this point with the appropriate notation (an open dot or a parenthesis, indicating that the point x = 2 wasn't included in the solution), and then you'd shade everything to the right, because "greater than" meant "everything off to the right". The steps for graphing two-variable linear inequalities are very much the same. Winpossible's online math courses and tutorials have gained rapidly popularity since their launch in 2008. Over 100,000 students have benefited from Winpossible's courses... these courses in conjunction with free unlimited homework help serve as a very effective math-tutor for our students. All of the Winpossible math tutorials have been designed by top-notch instructors and offer a comprehensive and rigorous math review of that topic. We guarantee that any student who studies with Winpossible, will get a firm grasp of the associated problem-solving techniques. Each course has our instructors providing step-by-step solutions to a wide variety of problems, completely demystifying the Winpossible courses have been used by students for help with homework and by homeschoolers. Several teachers use Winpossible courses at schools as a supplement for in-class instruction. They also use our course structure to develop course worksheets.

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The Norwegian Fjords are steep, ice-carved valleys that stretch from the land out into the sea. Fjords are created not solely by glacier erosion, but also by the high-pressure melt water that flows beneath the ice. Fjord valleys can be carved hundreds to thousands of meters below sea level. The Hardangerfjorden shown in this image is about 179 km long, and reaches its maximum depth of more than 800 m about 100 km inland. In the above image, based on elevation data collected by the Shuttle Radar Topography Mission (SRTM), beige and yellow represent low elevations, while red, brown and white represent progressively higher elevations. Shades of blue represent water. The Hardangerfjord is left of center, and extends off the top of the image. Sorfjorden is towards the right edge of the image. The location of a fjord may be due to pre-glacial valleys, bedrock characteristics, or fractures in the Earth's crust. Fjords typically have U-shaped cross-sectional profiles, with the valley floor being flat or only slightly rounded. The fjord's longitudinal profile usually consists of a series of basins separated by rock barriers or moraine sills (glacial debris). Fjord entrances are usually quite shallow with shoals and small islands. Usually the deep basins are situated some distance inland from the mouth of the fjord. The shallow mouths are places where the glaciers that once filled the valley either began to float, or else had room to spread out. Inland, the glaciers were more confined, and so they carved more deeply into the Earth. About 10,000 years ago, at the end of the last major glaciation, the Scandinavian land mass began slowly rising up as warmer temperatures freed it from the enormous weight of glacial ice, a process called glacial rebound. However, the land's increased buoyancy did not keep pace with the rising sea level, and the lower parts of formerly glaciated valleys became flooded. The glacial rebound of the Scandinavian land mass is still occurring.

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- To comprehend and respond to books read aloud - To understand the concept of characters - To respond to questions - To use expressive language - To recall and retell parts of a selection - To build vocabulary Read Corduroy by Don Freeman, or another book with interesting characters. Tell children that people, animals, and talking toys in stories are called characters. Hold up a book you've read recently in class, and ask children to name some of its characters. - Read the new book. Then page back through it and ask children: Who are the characters in this book? Invite children to describe the characters. - Ask children: Which character do you like best? Why? What did that character do in the story? Help each child participate, even if it is to repeat another child's response. Help children order the sequence of events in the book. Ask: What happened first? Next? Last? - Proficient - Child listens attentively to the story and is able to name and make accurate observations about the characters. - In Process - Child listens fairly attentively to the story and shows understanding by actions, such as laughing or pointing, but needs prompting to name or describe characters. - Not Yet Ready - Child is distracted and does not yet show an understanding of the story characters. More on: Activities for Preschoolers Excerpted from School Readiness Activity Cards. The Preschool Activity Cards provide engaging and purposeful experiences that develop language, literacy, and math skills for preschool children.

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Solve equations that require two operations to isolate the variable. A list of student-submitted discussion questions for Two-Step Equations and Properties of Equality. To encourage students’ critical thinking about vocabulary concepts, to allow students to reflect on their knowledge of individual vocabulary words, and to increase vocabulary comprehension using the Vocabulary Self-Rate. Come up with questions about a topic and learn new vocabulary to determine answers using the table To activate prior knowledge, to generate questions about a given topic, and to organize knowledge using a KWL Chart. Develop understanding of concepts by studying them in a relational manner. Analyze and refine the concept by summarizing the main idea, creating visual aids, and generating questions and comments using a Four Square Concept Matrix. Students will apply their understanding of solving two-step equations to learn how to finance a car over a period of 60 months. Find out why 30 degrees in the United States does not feel the same as 30 degrees in Italy. This study guide looks at the properties of equality and solving linear equations in one variable. It also looks at the number of solutions to linear equations in one variable.

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Before defining the surface area, I want to start from the bottom of this concept. Kids start to learn about points in grade five or six. Points can be compared to real numbers. For example, the dot we use to mark the period (full stop ".") is a point. A point can be represented by a capital letter. Following are two points A and B in space: A . . B When we put trillions of points side by side we get a line segment. Hence points give birth to a line. Now if we join the above two points we get a line AB. Hence, a line segment can be represented using two capital letters. There are many examples of lines in daily life, such as a sewing thread, an electric wire can be compared to a line. As billions of points construct a line, similarly many lines on a piece of paper or on land can give rise to the concept of area. Lines has only one way to go at a time, which means a line can only be top to bottom (vertical) or left to right (horizontal) but it can't be both. Hence lines have only one dimension and we call it "the length". As we measure other quantities, similarly, we can define units to measure the length of a line. We can measure the length of lines in metres, centimeters, millimeters, inches, yards or even in miles or kilometers. Below are some examples of different lines: Look at above lines, line segments, right angles, rays and also a spider web containing a bunch of lines and curves. Hope it has cleared the idea of dimensions in kids' minds. If kids get the idea of dimensions, and know that lines are only one dimensional then they are ready to learn area and then surface area. Now, if we draw two parallel lines vertically and two horizontally and let them intersect we get a very simple geometric shape called a rectangle as shown below: So, four lines when cut each other in a specific order, they make a rectangle. Look at above rectangle, it can be measured two ways, left to right (horizontal line called length) and top to bottom (vertical line called breadth or width). Therefore a rectangle have two dimensions called length and breadth (width). A page of a note book or textbook is the simplest example of a rectangle. Below are some more example of two dimensional shapes: Hence the lines give birth to a shape. Remember we are talking about straight lines. A curve is also a line but not straight. Kids can draw more shapes using lines. Here I want to stop; when lines make a two dimensional shape, they occupy some space on the surface over where they have drawn. This space bounded between lines is called "the area" or "surface area" in case of solid three dimensional shapes.

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Learning About Simple Sentences This is an introductory lesson about simple sentences. • to introduce the concept of simple sentences; to review the sentence parts, subject and predicate; to reinforce the idea that sentences make a complete thought. You have learned that a sentence is a group of words that makes complete sense. All sentences must have two things: 1. A Subject: who or what does the action. 2. A Predicate: what the subject does. Subjects are nouns or pronouns. A sentence with one subject and one predicate is called a simple sentence. For example: ? Jimmy built his own rocket. ? We watched him. The predicate is the action word. Predicates are always verbs.

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Geometry for Elementary School/Pythagorean theorem In this chapter, we will discuss the Pythagorean theorem. It is used the find the side lengths of right triangles. It says: - In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse). This means that if is a right triangle, the length of the hypotenuse, c, squared eqauls the sum of a squared plus b squared. Or: Here's an example: In a right-angled triangle, a=5cm and b=12cm, so what is c? If c is not larger than a or b, your answer is incorrect. There may be a number of reasons that your answer is incorrect. The first is that you have calculated the sums wrong, the second is that the triangle you are trying to find the hypotenuse of is not a right angled triangle or the third is you have mixed up the measurements. There may be more finer points to having a wrong answer but the three stated are the most common

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Introduction to Rational Functions - Concept Once we have a thorough understanding of polynomials we can look at rational functions that are a quotient of two polynomials. These rational functions have certain behaviors, and students are often asked to find their limits, or to graph them. Their graphs can have different characteristics depending on whether the numerator function has degree less than, equal to, or greater than the denominator function. I want to talk about a very important class of functions called rational functions. A rational function is one that can be written f of x equals p of x over q of x where p of x and q of x are polynomials. Now, f of x is defined for any number of x unless q of x the denominator equals zero so the domain will be all real numbers except those that make the denominator zero. And the zeros of a rational function will be the zeros of the numerator just as long as they are not also zeros of the denominator, so let's practice using these definitions in an example. Each of these three is a rational function, polynomial divided by polynomial so p of x over q of x. Now, find the domain and zeros. The domain of this function is going to be all real numbers except where the denominator is zero, so where is the denominator zero? 2x-5=0 when 2x=5, so we divide by 2, x equals five halves so the domain is all real numbers except five halves, all real numbers except five halves, now what are the zeros? For the zeros we look to the numerator. When is the numerator equal to zero? 2x squared minus 5x minus 3. Now this looks like it's factorable so I'm going to try to factor it 2x, x. I need a 3 and a 1 now if I put -3 here and +1 here I'll get x-6x, -5x that works. That means that x equals negative one half and x=3 are both zeros of this function and because neither of those zeros is also a zero of the denominator, these are going to be zeros of my function so the zeros are negative one half, x=3. Okay let's take a look at this guy, what's the domain? Well first we have to figure out where the where the denominator equals zero, so x squared minus 4x equal 0, I can factor this it equal zero when x is 0 or 4, so the domain will be all real numbers except 0 or 4, all real numbers except 0 or 4, now for the zeros of the function the numbers that make this function 0 we look to the numerator, x squared minus 1 equals zero and that's really easy x squared equals 1, x equals plus or minus 1, so as long as plus or minus 1 are not also zeros of the denominator, these are zeros of my function so the zeros are plus and minus 1. Finally let's look look at this function, this denominator I can find the zeros by factoring, x cubed minus x squared minus 6x equals 0, so you get x times x squared minus x minus 6 and this can also be factored looks like it's going to be x and x. I need maybe a 2 and a 3 if I go -3+2 I get my minus 6 and I get -3x+2x negative x that works, so the zeros of the denominator are x=0, 3 or -2, so the domain will be all real numbers except those three. Domain all reals except 0, 3 or -2. And then what about the zeros of this function? Let's look at the numerator; x squared minus 4 equals zero means x squared equals 4 so x is plus or minus 2. Now here's a case where one of the zeros of the numerator is also a zero of the denominator now because 2 is a zero of both the numerator and denominator, the function is not going to be defined there so you can't say that the function's value is 0 there, its not a 0 the only 0 will be then be -2 again I'm sorry actually -2 is this is the it's where it's undefined so positive 2. Let me just clarify, the function is not defined at -2 so -2 can't be a zero so it has to be +2 only.

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Although the surface is cold, the base of an ice sheet is generally warmer due to geothermal heat. In places, melting occurs and the melt-water lubricates the ice sheet so that it flows more rapidly. This process produces fast-flowing channels in the ice sheet — these are ice streams. The present-day polar ice sheets are relatively young in geological terms. The Antarctic Ice Sheet first formed as a small ice cap (maybe several) in the early Oligocene, but retreating and advancing many times until the Pliocene, when it came to occupy almost all of Antarctica. The Greenland ice sheet did not develop at all until the late Pliocene, but apparently developed very rapidly with the first continental glaciation. This had the unusual effect of allowing fossils of plants that once grew on present-day Greenland to be much better preserved than with the slowly forming Antarctic ice sheet. The Antarctic ice sheet is the largest single mass of ice on Earth. It covers an area of almost 14 million km² and contains 30 million km³ of ice. Around 90% of the fresh water on the Earth's surface is held in the ice sheet, and, if melted, would cause sea levels to rise by 61.1 meters. The Antarctic ice sheet is divided by the Transantarctic Mountains into two unequal sections called the East Antarctic ice sheet (EAIS) and the smaller West Antarctic Ice Sheet (WAIS). The EAIS rests on a major land mass but the bed of the WAIS is, in places, more than 2,500 meters below sea level. It would be seabed if the ice sheet were not there. The WAIS is classified as a marine-based ice sheet, meaning that its bed lies below sea level and its edges flow into floating ice shelves. The WAIS is bounded by the Ross Ice Shelf, the Ronne Ice Shelf, and outlet glaciers that drain into the Amundsen Sea. The Greenland ice sheet occupies about 82% of the surface of Greenland, and if melted would cause sea levels to rise by 7.2 metres. Estimated changes in the mass of Greenland's ice sheet suggest it is melting at a rate of about 239 cubic kilometres (57.3 cubic miles) per year. These measurements came from NASA's Gravity Recovery and Climate Experiment (GRACE) satellite, launched in 2002, as reported by BBC News in August 2006 . The IPCC projects that ice mass loss from melting of the Greenland ice sheet will continue to outpace accumulation of snowfall. Accumulation of snowfall on the Antarctic ice sheet is projected to outpace losses from melting. However, loss of mass on the Antarctic sheet may continue, if there is sufficient loss to outlet glaciers. According to the IPCC, understanding of dynamic ice flow processes is "limited".

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Rationale: Children need alphabetic insight about letters and phonemes to help them with reading skills. Before children can understand that letters and phonemes match, they have to be able to hear phonemes in spoken words. This lesson will help children hear that p says /p/. They will learn to hear /p/ in spoken words. Materials: Primary paper, pencils, chart with Put the pretty pumpkin onto the perfect platform, cards with pictures on them (see = #6), classroom board for example in #3, pictures of a pirate, a pumpkin, a boat, a clown, and a princess. Procedures: 1. Introduce the lesson with an explanation of alphabetic code and phonemes. Explain that "today we will be learning about p and the sound it makes, /p/." Tell the students, "/p/ sounds kind of like popcorn when it is popping in the microwave. Let's see if you can make the /p/ sound like popping popcorn." Have the students pop up out of their seats when they say or hear /p/. 2. Next, say the tongue twister that has p's in it from the chart. Have the students pop up like popcorn when they hear the /p/ sound. Put the pretty pumpkin onto the perfect platform. Have the students repeat it after you. Then say it one more time. Next, say it again and stretch out the /p/ in each word. Model to the students and then have them do it themselves. Then have the students do it alone. 3. Explain to the students that now that they know what p sounds like they are going to learn what it looks like and how to write it. Pass out primary paper and model on the board how to write a p. Be sure to draw lines on the board like primary paper and walk them through the process. Have the students try it while still explaining. Have them write multiple upper and lowercase p's on their papers. 4. Tell the students, "Now I'm going to say a list of words and I want you to pop up like popcorn when you hear /p/." Say, princess, pineapple, motorcycle, popcorn, bird, pie, captain, pumpkin, pig, and owl. "Drag out the /p/ sound in the p words. 5. Show the students pictures of a pirate, a pumpkin, a boat, a clown, and a princess. Have them pop up like popcorn when they see a picture with the sound /p/ in it. 6. For an assessment, pass out cards with two pictures each on them, one with a /p/ sound and one without. Have each student circle which picture represents /p/. Murray, Bruce. "Example of Emergent Literacy Design: Sound the Foghorn". Harris, Katherine. "Penelope, The Precious Pig". http://www.auburn.edu/academic/education/reading_genie/voyages/harrisel.html Return to Passages Index

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Students will learn how to graph motion vs time. Students will learn how to take the slope of a graph and relate it to the instantaneous velocity or acceleration for position or velocity graphs. Finally students will learn how to take the area of a velocity vs time graph in order to calculate the displacement. Illustrates the structure of position-time graphs with positive acceleration and explains why they are shaped this way. Illustrates the structure of position-time graphs with negative acceleration and explains why they are shaped this way. Illustrates velocity-time graphs and how acceleration affects them with an example problem. Know the significance of slope, curve, and direction on a position-time graph and a velocity-time graph, sketch the graph of a moving object given a description of its motion. A list of student-submitted discussion questions for Graphing Motion. A review of the terms distance, displacement, velocity, speed, and acceleration. Also looks at graphs of motion, using the kinematic equations, projectile motion, and free fall.

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This is a drawing of a process which forms mountains on Earth. Click on image for full size Mountains are built through a general process called "deformation" of the crust of the Earth. Deformation is a fancy word which could also mean "folding". An example of this kind of folding comes from the process described below. When two sections of the Earth's lithosphere collide, rather than being subducted, where one slab of lithosphere is forced down to deeper regions of the Earth, the slabs pile into each other, causing one or both slabs can fold up like an accordion. This process elevates the crust, folds and deforms it heavily, and produces a mountain range. Mountain building and mantle subduction usually go together. This process is illustrated in the figure to the left. The lithospheric slab on the right is subducted, while the force of the collision gradually causes the slab on the left to fold deeply. Along the way, melting of the subducted slab leads to volcano formation. Shop Windows to the Universe Science Store! You might also be interested in: When two sections of the Earth's crust collide, one slab of crust can be forced back down into the deeper regions of the Earth, as shown in this diagram. This process is called subduction. The slab that...more Volcanoes form when hot material from below rises and leaks into the crust. This hot material, called magma, comes either from a melt of subducted crustal material, and which is light and buoyant after...more Many kinds of surface features are clues that our lithosphere is sliding. Two types of features can form when plates move apart. At mid ocean ridges, the bottom of the sea splits apart and new crust is...more Peru's new president, Alejandro Toledo, was sworn in on July 28, 2001. The next day he visited Machu Picchu and Cuzco. There were ceremonies at both locations where Toledo asked for the blessing of the...more Instead of icy-volcanism, the surface of Ganymede reveals gradual surface deformation and stretching, a stretching similar to the crustal deformation of the Earth. In this case, the folding and stretching...more Ash is made of millions of tiny fragments of rock and glass formed during a volcanic eruption. Volcanic ash particles are less than 2 mm in size and can be much smaller. Volcanic ash forms in several ways...more Cinder cones are simple volcanoes which have a bowl-shaped crater at the summit and only grow to about a thousand feet, the size of a hill. They usually are created of eruptions from a single opening,...more

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Why did the US Congress have a problem in 1850? And why did the solution lead to the creation, 160 years ago this month, of a place called Utah? The lands of the American Southwest – an area now covering California, Wyoming, Colorado, New Mexico, Arizona, Nevada, and Utah – were ceded to the United States following the end of the Mexican-American War in 1848. The problem confronting the US, however, was whether the new lands should become slave states or free. The union of the nation depended on keeping a balance, and for two years, Congress wrestled with the question. In the absence of any decision, people living in those areas began to organize governing institutions of their own. Mormon leader Brigham Young established an independent government called Deseret, which stretched from the Rocky Mountains to the sea, and petitioned for statehood as a way to secure local independence. Finally, members of Congress made a deal called the Compromise of 1850, wherein California was admitted to the Union as a free state, while the remaining lands became the New Mexico and Utah Territories, where people were allowed to decide the slavery issue for themselves. The compromise resolved the immediate crisis, but only delayed the question of slavery in western lands. Congress also refused to grant statehood to Deseret because the region lacked the required number of eligible voters. Moreover, they objected to the huge size of the proposed state. When selecting a name for the new territory, Congressional support was strong for the name Utah, after the indigenous Ute tribe. Mormons resisted naming the territory after a people they scorned and feared, but the name prevailed. So, in September 1850, Congress passed a bill organizing the Utah Territory, rejecting the name Deseret and shrinking its presumptuous borders. However, President Millard Fillmore's politically astute selection of Brigham Young as governor made territorial status easier for the Mormons to accept. In gratitude, they named their new territorial capital and its surrounding county after him. Utah would wait another 46 years for statehood. Beehive Archive is a production of the Utah Humanities Council. Sources consulted in the creation of the Beehive Archive and past episodes may be found at www.utahhumanities.org/BeehiveArchive.htm

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Solon was one of the archons in ancient Athens. In 594 B.C. Solon made several important reforms, which loosened the tension of civil war breaking out. He also made it were the public office system was based on wealth. Thus making it were any qualified citizen could become a public official.Solon also published all the laws of the Athenian Society. Solons reforms, even though important, did not solve the problem of poverty in Athens. Because of this, Pisistratus was able to sieze power and become a tyrant. He ruled from 545 B.C. to 527 B.C. He did though continue the work of Solon by reducing the power of the traditional ruling class. Cleisthenes was the founder of democracy in Athens. He proposed the constitution in 508 B.C. The constitution made Athens a democracy. The curious thing about the constitution was that it stayed intact for several hundred years. This may not seem strange, but it was because the constitution was unwritten. The ideaology was based on Solon, but it also provided conditions that greatly developed them. The new constitution now made where all men of 18 years or older were registered as citizens and were members of the village which they lived in. Which gave each person a vote in the society. The constitution made were 500 of the people made the decisions in the city and those officials were elected each year. Each citizen had a chance to run the city. Women were not considered citizens thus they could not vote. People can be banished under this system for 10 years by a majority vote of the populis.

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Algebraic concepts can evolve and continue to develop during prekindergarten through grade 2. They will be manifested through work with classification, patterns and relations, operations with whole numbers, explorations of function, and step-by-step processes. Although the concepts discussed in this Standard are algebraic, this does not mean that students in the early grades are going to deal with the symbolism often taught in a traditional high school algebra course. Even before formal schooling, children develop beginning concepts related to patterns, functions, and algebra. They learn repetitive songs, rhythmic chants, and predictive poems that are based on repeating and growing patterns. The recognition, comparison, and analysis of patterns are important components of a student's intellectual development. When students notice that operations seem to have particular properties, they are beginning to think algebraically. For example, they realize that changing the order in which two numbers are added does not change the result or that adding zero to a number leaves that number unchanged. Students' observations and discussions of how quantities relate to one another lead to initial experiences with function relationships, and their representations of mathematical situations using concrete objects, pictures, and symbols are the beginnings of mathematical modeling. Many of the step-by-step processes that students use form the basis of understanding iteration and recursion. and ordering facilitate work with patterns, geometric shapes, and data. Given a package of assorted stickers, children quickly notice many differences among the items. They can sort the stickers into groups having similar traits such as color, size, or design and order them from smallest to largest. Caregivers and teachers should elicit from children the criteria they are using as they sort and group objects. Patterns are a way for young students to recognize order and to organize their world and are important in all aspects of mathematics at this level. Preschoolers recognize patterns in their environment and, through experiences in school, should become more skilled in noticing patterns in arrangements of objects, shapes, and numbers and in using patterns to predict what comes next in an arrangement. Students know, for example, that "first comes breakfast, then school," and "Monday we go to art, Tuesday we go to music." Students who see the digits "0, 1, 2, 3, 4, 5, 6, 7, 8, 9" repeated over and over will see a pattern that helps them learn to count to 100a formidable task for students who do not recognize the pattern. Teachers should help students develop the ability to form generalizations by asking such questions as "How could you describe this pattern?" or "How can it be repeated or extended?" or "How are these patterns alike?" For example, students should recognize that the color pattern "blue, blue, red, blue, blue, red" is the same in form as "clap, clap, step, clap, clap, step." This recognition lays the foundation for the idea that two very different situations can have the same mathematical » features and thus are the same in some important ways. Knowing that each pattern above could be described as having the form AABAAB is for students an early introduction to the power of algebra. By encouraging students to explore and model relationships using language and notation that is meaningful for them, teachers can help students see different relationships and make conjectures and generalizations from their experiences with numbers. Teachers can, for instance, deepen students' understanding of numbers by asking them to model the same quantity in many waysfor example, eighteen is nine groups of two, 1 ten and 8 ones, three groups of six, or six groups of three. Pairing counting numbers with a repeating pattern of objects can create a function (see fig. 4.7) that teachers can explore with students: What is the second shape? To continue the pattern, what shape comes next? What number comes next when you are counting? What do you notice about the numbers that are beneath the triangles? What shape would 14 be? Students should learn to solve problems by identifying specific processes. For example, when students are skip-counting three, six, nine, twelve, ..., one way to obtain the next term is to add three to the previous number. Students can use a similar process to compute how much to pay for seven balloons if one balloon costs 20¢. If they recognize the sequence 20, 40, 60, ... and continue to add 20, they can find the cost for seven balloons. Alternatively, students can realize that the total amount to be paid is determined by the number of balloons bought and find a way to compute the total directly. Teachers in grades 1 and 2 should provide experiences for students to learn to use charts and tables for recording and organizing information in varying formats (see figs. 4.8 and 4.9). They also should discuss the different notations for showing amounts of money. (One balloon costs 20¢, or $0.20, and seven balloons cost $1.40.) Two central themes of algebraic thinking are appropriate for young students. The first involves making generalizations and using symbols to represent mathematical ideas, and the second is representing and solving problems (Carpenter and Levi 1999). For example, adding pairs of numbers in different orders such as 3 + 5 and 5 + 3 can lead students to infer that when two numbers are added, the order does not matter. As students generalize from observations about number and operations, they are forming the basis of algebraic thinking. Similarly, when students decompose numbers in order to compute, they often use the associative property for the computation. For instance, they may compute 8 + 5, saying, "8 + 2 is 10, and 3 more is 13." Students often discover and make generalizations about other properties. Although it is not necessary to introduce vocabulary such as commutativity or associativity, teachers must be aware of the algebraic properties used by students at this age. They should build students' understanding of the importance of their observations about mathematical situations and challenge them to investigate whether specific observations and conjectures hold for all cases. Teachers should take advantage of their observations of students, as illustrated in this story drawn from an experience in a kindergarten class. The teacher had prepared two groups of cards for her students. In the first group, the number on the front and back of each card differed by 1. In the second group, these numbers differed by 2. | The teacher showed the students a card with 12 written on it and explained, "On the back of this card, I've written another number." She turned the card over to show the number 13. Then she showed the students a second card with 15 on the front and 16 on the back. » As she continued showing the students the cards, each time she asked the students, "What do you think will be on the back?" Soon the students figured out that she was adding 1 to the number on the front to get the number on the back of the card. Then the teacher brought out a second set of cards. These were also numbered front and back, but the numbers differed by 2, for example, 33 and 35, 46 and 48, 22 and 24. Again, the teacher showed the students a sample card and continued with other cards, encouraging them to predict what number was on the back of each card. Soon the students figured out that the numbers on the backs of the cards were 2 more than the numbers on the fronts. When the set of cards was exhausted, the students wanted to play again. "But," said the teacher, "we can't do that until I make another set of cards." One student spoke up, "You don't have to do that, we can just flip the cards over. The cards will all be minus 2." As a follow-up to the discussion, this teacher could have described what was on each group of cards in a more algebraic manner. The numbers on the backs of the cards in the first group could be named as "front number plus 1" and the second as "front number plus 2." Following the student's suggestion, if the cards in the second group were flipped over, the numbers on the backs could then be described as "front number minus 2." Such activities, together with the discussions and analysis that follow them, build a foundation for understanding the inverse relationship. Through classroom discussions of different representations during the pre-K2 years, students should develop an increased ability to use symbols as a means of recording their thinking. In the earliest years, teachers may provide scaffolding for students by writing for them until they have the ability to record their ideas. Original representations remain important throughout the students' mathematical study and should be encouraged. Symbolic representation and manipulation should be embedded in instructional experiences as another vehicle for understanding and making sense of mathematics. Equality is an important algebraic concept that students must encounter and begin to understand in the lower grades. A common explanation of the equals sign given by students is that "the answer is coming," but they need to recognize that the equals sign indicates a relationshipthat the quantities on each side are equivalent, for example, 10 = 4 + 6 or 4 + 6 = 5 + 5. In the later years of this grade band, teachers should provide opportunities for students to make connections from symbolic notation to the representation of the equation. For example, if a student records the addition of four 7s as shown on the left in figure 4.11, the teacher could show a series of additions correctly, as shown on the right, and use a balance and cubes to demonstrate the equalities. » Students should learn to make models to represent and solve problems. For example, a teacher may pose the following problem: There are six chairs and stools. The chairs have four legs and the stools have three legs. All together there are twenty legs. How many chairs and how many stools are there? One student may represent the situation by drawing six circles and then putting tallies inside to represent the number of legs. Another student may represent the situation by using symbols, making a first guess that the number of stools and chairs is the same and adding 3 + 3 + 3 + 4 + 4 + 4. Realizing that the sum is too large, the student might adjust the number of chairs and stools so that the sum of their legs is 20. Change is an important idea that students encounter early on. When students measure something over time, they can describe change both qualitatively (e.g., "Today is colder than yesterday") and quantitatively (e.g., "I am two inches taller than I was a year ago"). Some changes are predictable. For instance, students grow taller, not shorter, as they get older. The understanding that most things change over time, that many such changes can be described mathematically, and that many changes are predictable helps lay a foundation for applying mathematics to other fields and for understanding the world. |Home | Table of Contents | Purchase | Resources| |NCTM Home | Illuminations Web site| Copyright © 2000 by the National Council of Teachers of Mathematics.

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A < B A > B A <= B A >= B A == B A ~= B The relational operators are <, >, <=, >=, ==, and ~=. Relational operators perform element-by-element comparisons between two arrays. They return a logical array of the same size, with elements set to logical 1 (true) where the relation is true, and elements set to logical 0 (false) where it is not. The operators <, >, <=, and >= use only the real part of their operands for the comparison. The operators == and ~= test real and imaginary parts. To test if two strings are equivalent, use strcmp, which allows vectors of dissimilar length to be compared. Note For some toolboxes, the relational operators are overloaded, that is, they perform differently in the context of that toolbox. To see the toolboxes that overload a given operator, type help followed by the operator name. For example, type help lt. The toolboxes that overload lt (<) are listed. For information about using the operator in that toolbox, see the documentation for the toolbox. If one of the operands is a scalar and the other a matrix, the scalar expands to the size of the matrix. For example, the two pairs of statements X = 5; X >= [1 2 3; 4 5 6; 7 8 10] X = 5*ones(3,3); X >= [1 2 3; 4 5 6; 7 8 10] produce the same result: ans = 1 1 1 1 1 0 0 0 0

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