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From Wikipedia
In the United States the Singapore Math Method of teaching mathematics is based on the primary textbooks and syllabus from the national curriculum of Singapore. These textbooks have a consistent and strong emphasis on problem solving and model drawing, with a focus on indepth understanding of the essential math skills recommended in the NCTM Curriculum Focal Points (National Council of Teachers of Mathematics), the National Mathematics Advisory Panel, and the proposed Common Core State Standards.
Explanations of math concepts are exceptionally clear and simple (often just a few words in a cartoon balloon), so that Englishasasecond language students (common in Singapore) can read it easily. The method has become more popular since the release of scores from the Trends in International Mathematics and Science Study in 2003 showed Singapore at the top of the world in 4th and 8th grade mathematics. This was the third study by the NCES, and the 2007 TIMSS was released in December 2008.
Currently, there are three comprehensive Singapore Math series adapted for the United States (US currency and units of measure, plus extra math topics that are currently popular in state math standards) and available through USbased companies. Primary Mathematics US Edition (used in the US since 2003) was the first effort to adapt Singaporeâ€™s Primary Mathematics to the US. Primary Mathematics Standards Edition is a newer variant, adopted by the California Department of Education in 2007. Math in Focus: the Singapore Approach (available since 2009) is the American edition of Singaporebased publisher Marshall Cavendishâ€™s My Pals Are Here! Maths, a textbook series that is currently used by over 80% of schools in Singapore. The first two are available from Oregonbased SingaporeMath.com. The third is published by Great Source, an imprint of Bostonbased Houghton Mifflin Harcourt.
History
Prior to 1980, Singapore imported all of its mathematics textbooks from other nations. Beginning in 1980, however, Singapore began to take a new approach to mathematics instruction. Instead of importing its mathematics textbooks, the Curriculum Development Institute of Singapore (CDIS) was established. One charge of CDIS was to develop primary and secondary textbooks. At the same time, the Ministry of Education, the centralized education authority in the country, set new goals for mathematics education. These goals emphasized a focus on problem solving and on heuristic model drawing. The CDIS incorporated these goals into the textbooks, and in 1982 the first Singapore math program, Primary Mathematics 16, was published. In 1992, a second edition was made available. The second edition revisions included an even stronger focus on problem solving and on using model drawing as a strategy to problem solve.
The country continued to develop its mathematics program. Further revisions included:
 Creating a tighter content focus of the mathematics curricula following a study to review the scope and sequence in 1998
 Privatizing the production of the primary level mathematics textbooks in 2001, with the hope that collaboration among textbook publishers would lead to quality textbooks at more affordable prices
 Placing an even greater focus on developing mathematical concepts and fostering mathematical problem solving in 2006 revisions
Following Singaporeâ€™s curricular and instructional initiatives, dramatic improvements in math proficiency for Singapore students on international assessments were seen. In 1984, Singaporeâ€™s students were placed 16th out of 26 nations in the Second International Science Study (SISS). By 1995, the Trends in International Mathematics and Science Study (TIMSS) ranked Singaporeâ€™s students first among participating nations. The 2007 results also showed Singapore as a topperforming nation.
Features
 Each semesterlevel Singapore Math textbook builds upon preceding levels, and assumes that what was taught need not be taught again. Consequently, it is necessary to assign Singapore Math students to a textbook that matches what they are ready to learn next. (Placement exams are available online.) By contrast, the typical US classroom offers the same gradelevel math instruction to all students, reviews previously taught math skills before teaching new skills, and gives more emphasis to topics that donâ€™t build on previously taught math skills (bar graphs, geometric shapes, measurement units).
 A great deal of instructional time is saved by focusing on essential math skills, and by not reteaching what has been taught before. In fact, some teachers report that Singapore Math feels slower paced than what theyâ€™re used to. However, the result is that students master essential math skills at a more rapid pace. By the end of sixth grade, Singapore Math students have mastered multiplication and division of fractions, and they are comfortable doing difficult multistep word problems. With that foundation, they are well prepared to complete Algebra 1 in middle school.
 Singapore Math students begin solving simple multistep word problems in third grade, using a technique called the â€œbar modelâ€� method. Later grades apply this same method to more and more difficult problems, so that by sixth grade they are solving very difficult problems like this: â€œLauren spent 20 percent of her money on a dress. She spent 2/5 of the remainder on a book. She had $72 left. How much money did she have at first?â€� Consequently, when a school first adopts Singapore Math, the upper elementary grades will need to catch up on what they missed. This can be done by going through the problemsolving chapters in the preceding grade levels, or by using a Singapore Math Model Method supplemental textbook.
 The principle of teaching mathematical concepts from concrete through pictorial to abstract. For example, introduction of abstract decimal fractions (in Grade 4) is preceded by their pictorial model of centimeters and millimeters on a metric ruler, but even earlier (in Grades 2 and 3) addition and subtraction of decimals is studied in the concrete form of dollars and cents.
 Systematic use of word problems as the way of building the semantics of mathematical operations. Simply put, students learn when to add and when to subtract, relying on the meaning of the situation (rather than "cluewords," as often done in the US schools). Formulations are free of any redundancies, and challenge students' understanding of mathematics only. This is different from many U.S. curricula, where word problems are to show "applications" of math and are spiced with immaterial details intended to obscure the mathematical content of the problem.
 The need for repetitive drill is minimized by clever sequencing of the topics. For instance, the introduction of multiplication facts by 2, 3, 4 and 5 in the middle of Grade 2 is followed by a seemingly unrelated section on reading statistical data from a graph. In fact, the latter task reinforces the learning of multiplication facts when the scale begins to vary from 2 to 5 objects per graphical unit.
 The use of barmodels in teaching problem solving (a form of prealgebra). This device is as old as Book V of Euclid's Elements, written in the 4th century BC, and consists simply in representing (mentally or graphically) arithmetical quantities by line segments. In SM books, such line segments are regularly used to show and teach one's thi
Saxon math, developed by John Saxon, is a teaching method for incremental learning of mathematics. It involves teaching a new mathematical concept every day and constantly reviewing old concepts. Early editions were deprecated for providing very few opportunities to practice the new material before plunging into a review of all previous material. Newer editions typically split the day's work evenly between practicing the new material and reviewing old material. Its primary strength is in a steady review of all previous material, which is especially important to students who struggle with retaining the math they previously learned.
In all books before Algebra 1/2 (the equivalent of a PreAlgebra book), the book is designed for the student to complete assorted mental math problems, learn a new mathematical concept, practice problems relating to that lesson, and solve a varied number of problems which include what the students learned today and in select previous lessons  all for one day's class. This daily cycle is interrupted for tests and additional topics. In the Algebra 1/2 book and all higher books in the series, the mental math is dropped, and tests are given more frequently.
The Saxon math program has a specific set of products to support homeschoolers, including solution keys and readymade tests, which makes it popular among some homeschool families. It has also been adopted as an alternative to reform mathematics programs in public and private schools. Saxon teaches familiar algorithms and uses familiar terminology, unlike many reform texts, which also contributes to its popularity.
Replacing standardsbased texts
By the mid 2000s, many school districts were considering abandoning experiments with reform approaches which had not produced acceptable test scores. For example, school board member Debbie Winskill in Tacoma, Washington said that the nontraditional Interactive Mathematics Program (IMP) "has been a dismal failure." Speaking to the board, Mount Tahoma High School teacher Clifford Harris noted that he taught sophomores in another district Saxon Math, and their Washington Assessment of Student Learning scores have continually climbed. Unlike IMP, Saxon program gives students plenty of chances to review material so they retain their skills, he said. In September 2006, Tacoma Public Schools introduced the Saxon books districtwide and rejected the previous IMP textbooks.
Rule 30 is a onedimensional binary cellular automaton rule introduced by Stephen Wolfram in 1983. Wolfram describes it as being his "alltime favourite rule" and details it in his book, A New Kind of Science. UsingWolfram's classification scheme, Rule 30 is a Class III rule, displaying aperiodic, chaotic behaviour.
This rule is of particular interest because it produces complex, seeminglyrandom patterns from simple, welldefined rules. Because of this, Wolfram believes that rule 30, and cellular automata in general, are the key to understanding how simple rules produce complex structures and behaviour in nature. For instance, a pattern resembling Rule 30 appears on the shell of the widespread cone snail species Conus textile. Rule 30 has also been used as arandom number generator in Wolfram's program Mathematica, and has also been proposed as a possible stream cipher for use in cryptography. However, Sipper and Tomassini have shown that as a random number generator rule 30 exhibits poor behavior on a chi squared test compared to other cellular automaton based generators.
Rule 30 is so named because 30 is the smallest Wolfram code which describes its rule set (as described below). The mirror image, complement, and mirror complement of Rule 30 have Wolfram codes 86, 135, and 149, respectively.
Rule set
In all of Wolfram's elementary cellular automata, an infinite onedimensional array of cellular automaton cells with only two states is considered, with each cell in some initial state. At discrete time intervals, every cell spontaneously changes state based on its current state and the state of its two neighbors. For Rule 30, the rule set which governs the next state of the automaton is:
The following diagram shows the pattern created, with cells colored based on the previous state of their neighborhood. Darker colors represent "1" and lighter colors represent "0". Time increases down the vertical axis.
Structure and properties
The following pattern emerges from an initial state in a single cell with state 1 (shown as black) is surrounded by cells with state 0 (white).
Rule 30 cellular automaton
Here, the vertical axis represents time and any horizontal crosssection of the image represents the state of all the cells in the array at a specific point in the pattern's evolution. Several motifs are present in this structure, such as the frequent appearance of white triangles and a welldefined striped pattern on the left side; however the structure as a whole has no discernible pattern. The number of black cells at generation n is given by the sequence
 1, 3, 3, 6, 4, 9, 5, 12, 7, 12, 11, 14, 12, 19, 13, 22, 15, 19, ...
and is approximately n.
As is apparent from the image above, rule 30 generates seeming randomness despite the lack of anything that could reasonably be considered random input. Stephen Wolfram proposed using its center column as a pseudorandom number generator (PRNG); it passes many standard tests for randomness, and Wolfram uses this rule in the Mathematica product for creating random integers. Although Rule 30 produces randomness on many input patterns, there are also an infinite number of input patterns that result in repeating patterns. The trivial example of such a pattern is the input pattern only consisting of zeros. A less trivial example, found by Matthew Cook, is any input pattern consisting of infinite repetitions of the pattern '00001000111000', with repetitions optionally being separated by six ones. Many more such patterns were found by Frans Faase. See [http://www.iwriteiam.nl/Rule30.html Repeating Rule 30 patterns].
The slide rule, also known colloquially as a slipstick, is a mechanical analog computer. The slide rule is used primarily for multiplication and division, and also for functions such as roots, logarithms and trigonometry, but is not normally used for addition or subtraction.
Slide rules come in a diverse range of styles and generally appear in a linear or circular form with a standardized set of markings (scales) essential to performing mathematical computations. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in calculations common to that field.
William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier. Before the advent of the pocket calculator, it was the most commonly used calculation tool in science and engineering. The use of slide rules continued to grow through the 1950s and 1960s even as digital computing devices were being gradually introduced; but around 1974 the electronic scientific calculator made it largely obsolete and most suppliers left the business.
Basic concepts
In its most basic form, the slide rule uses two logarithmic scales to allow rapid multiplication and division of numbers. These common operations can be timeconsuming and errorprone when done on paper. More elaborate slide rules allow other calculations, such as square roots, exponentials, logarithms, and trigonometric functions.
Scales may be grouped in decades, which are numbers ranging from 1 to 10 (i.e. 10^{n} to 10^{n+1}). Thus single decade scales C and D range from 1 to 10 across the entire width of the slide rule while double decade scales A and B range from 1 to 100 over the width of the slide rule.
In general, mathematical calculations are performed by aligning a mark on the sliding central strip with a mark on one of the fixed strips, and then observing the relative positions of other marks on the strips. Numbers aligned with the marks give the approximate value of the product, quotient, or other calculated result.
The user determines the location of the decimal point in the result, based on mental estimation. Scientific notation is used to track the decimal point in more formal calculations. Addition and subtraction steps in a calculation are generally done mentally or on paper, not on the slide rule.
Most slide rules consist of three linear strips of the same length, aligned in parallel and interlocked so that the central strip can be moved lengthwise relative to the other two. The outer two strips are fixed so that their relative positions do not change.
Some slide rules ("duplex" models) have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip (which can usually be pulled out, flipped over and reinserted for convenience), still others on one side only ("simplex" rules). A sliding cursor with a vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales.
Operation
Multiplication
A logarithm transforms the operations of multiplication and division to addition and subtraction according to the rules \log(xy) = \log(x) + \log(y) and \log(x/y) = \log(x)  \log(y). Moving the top scale to the right by a distance of \log(x), by matching the beginning of the top scale with the label x on the bottom, aligns each number y, at position \log(y) on the top scale, with the number at position \log(x) + \log(y) on the bottom scale. Because \log(x) + \log(y) = \log(xy), this position on the bottom scale gives xy, the product of x and y. For example, to calculate 3Ã—2, the 1 on the top scale is moved to the 2 on the bottom scale. The answer, 6, is read off the bottom scale where 3 is on the top scale. In general, the 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top.
Operations may go "off the scale;" for example, the diagram above shows that the slide rule has not positioned the 7 on the upper scale above any number on the lower scale, so it does not give any answer for 2Ã—7. In such cases, the user may slide the upper scale to the left until its right index aligns with the 2, effectively multiplying by 0.2 instead of by 2, as in the illustration below:
Here the user of the slide rule must remember to adjust the decimal point appropriately to correct the final answer. We wanted to find 2Ã—7, but instead we calculated 0.2Ã—7=1.4. So the true answer is not 1.4 but 14. Resetting the slide is not the only way to handle multiplications that would result in offscale results, such as 2Ã—7; some other methods are:
 Use the doubledecade scales A and B.
 Use the folded scales. In this example, set the left 1 of C opposite the 2 of D. Move the cursor to 7 on CF, and read the result from DF.
 Use the CI inverted scale. Position the 7 on the CI scale above the 2 on the D scale, and then read the result off of the D scale, below the 1 on the CI scale. Since 1 occurs in two places on the CI scale, one of them will always be onscale.
 Use both the CI inverted scale and the C scale. Line up the 2 of CI with the 1 of D, and read the result from D, below the 7 on the C scale.
Method 1 is easy to understand, but entails a loss of precision. Method 3 has the advantage that it only involves two scales.
Division
The illustration below demonstrates the computation of 5.5/2. The 2 on the top scale is placed over the 5.5 on the bottom scale. The 1 on the top scale lies above the quotient, 2.75. There is more than one method for doing division
From Yahoo Answers
Answers:Discrete mathematics is the branch of mathematics dealing with objects that can assume only distinct, separated values. The term "discrete mathematics" is therefore used in contrast with "continuous mathematics," which is the branch of mathematics dealing with objects that can vary smoothly (and which includes, for example, calculus). Whereas discrete objects can often be characterized by integers, continuous objects require real numbers. The study of how discrete objects combine with one another and the probabilities of various outcomes is known as combinatorics. Other fields of mathematics that are considered to be part of discrete mathematics include graph theory and the theory of computation. Topics in number theory such as congruences and recurrence relations are also considered part of discrete mathematics. The study of topics in discrete mathematics usually includes the study of algorithms, their implementations, and efficiencies. Discrete mathematics is the mathematical language of computer science, and as such, its importance has increased dramatically in recent decades.  Example: ELECTION THEORY: Use preference ballots for voting methods: plurality, majority, runoff, sequential runoff, Borda Count and Condorcet. SOCIAL ISSUES: Study the weighted voting systems of Banzhaf and ShapleyShubik then compare to preferential voting. Conduct investigations using Fair Division Schemes: dividerchooser, lone divider, lone chooser, last diminisher, sealed bids, markers
Answers:ok so i think for this one its best if i go through the first one, and you do the rest by yourself. I will use (x,y) exists in p to mean "x is prelated to y" since you didnt say you knew what those things we're testing means, I'll go through them, ignore this if you want.  reflexivity means that all elements map to themselves. in your example this means that (x,x) exists in p, for all values of x. symmetry means that the relation always goes both ways, so in your example if (x,y) exists in p, so should (y,x) transitivity is a little harder to explain in words but in your example, if (x,y) exists in p and (y,z) exists in p, then so should (x,z). so for example, the = sign obeys this law, since if x=y, and y=z, then x=z. Antisymmetry is the opposite of symmetry! It means that distinct elements are never related to each other. so in your example, if (x,y) exists in p, (y,x) does not.  so first we test reflexivity. in other words, is there (x,x) in p, for all values of x? in your first example, we can see that this is true (1,1) (2,2) and (3,3) are all there. now symmetry, we look to see that all mirror images are there. (1,3) has (3,1) but unfortunately (2,3) is there, but (3,2) is not. transitivity might take a little longer with some of them, but in this one there are only a few things to check. (1,3) and (3,1) exist in p, so we need (1,1) to as well and it does. (here x=1, y=3 and z=1, for my explanation above) but (2,3) and (3,1) exist in p, and so we need (2,1), but it doesn't exist in p. so this one fails transitivity as well. ok and now we check antisymmetry. well we can see straight away that we have (1,3) and (3,1) and so it fails. hope that helps
Answers:I don't know the answer, but if it is a math problem, why is it in the Government section?
Answers:1. True  this is actually true for any undirected graph G (it need not be connected) 2. False  An Euler circuit needs to include all the edges. Just consider a regular polygon with a diagonal drawn. The polygon's edges is a circuit that includes every vertex, but it doesn't include the one diagonal, so it can't be an Euler circuit. 3. True  this is actually one way to define an Eulerian graph. By the way, I'm assuming that G is finite, since otherwise 1 and 3 may not even make any sense, and then even when they do, they could still be false when G is infinite.
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