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From Wikipedia
A pie chart (or a circle graph) is a circularchart divided into sectors, illustrating proportion. In a pie chart, the arc length of each sector (and consequently its central angle and area), is proportional to the quantity it represents. When angles are measured with 1 turn as unit then a number of percent is identified with the same number of centiturns. Together, the sectors create a full disk. It is named for its resemblance to a pie which has been sliced. The earliest known pie chart is generally credited to William Playfair's Statistical Breviary of 1801.
The pie chart is perhaps the most ubiquitous statistical chart in the business world and the mass media. However, it has been criticized, and some recommend avoiding it, pointing out in particular that it is difficult to compare different sections of a given pie chart, or to compare data across different pie charts. Pie charts can be an effective way of displaying information in some cases, in particular if the intent is to compare the size of a slice with the whole pie, rather than comparing the slices among them. Pie charts work particularly well when the slices represent 25 to 50% of the data, but in general, other plots such as the bar chart or the dot plot, or nongraphical methods such as tables, may be more adapted for representing certain information.It also shows the frequency within certain groups of information.
Example
The following example chart is based on preliminary results of the election for the European Parliament in 2004. The table lists the number of seats allocated to each party group, along with the derived percentage of the total that they each make up. The values in the last column, the derived central angle of each sector, is found by multiplying the percentage by 360Â°.
*Because of rounding, these totals do not add up to 100 and 360.
The size of each central angle is proportional to the size of the corresponding quantity, here the number of seats. Since the sum of the central angles has to be 360Â°, the central angle for a quantity that is a fraction Q of the total is 360Q degrees. In the example, the central angle for the largest group (European People's Party (EPP)) is 135.7Â° because 0.377 times 360, rounded to one decimal place(s), equals 135.7.
Use, effectiveness and visual perception
Pie charts are common in business and journalism, perhaps because they are perceived as being less "geeky" than other types of graph. However statisticians generally regard pie charts as a poor method of displaying information, and they are uncommon in scientific literature. One reason is that it is more difficult for comparisons to be made between the size of items in a chart when area is used instead of length and when different items are shown as different shapes. Stevens' power law states that visual area is perceived with a power of 0.7, compared to a power of 1.0 for length. This suggests that length is a better scale to use, since perceived differences would be linearly related to actual differences.
Further, in research performed at AT&T Bell Laboratories, it was shown that comparison by angle was less accurate than comparison by length. This can be illustrated with the diagram to the right, showing three pie charts, and, below each of them, the corresponding bar chart representing the same data. Most subjects have difficulty ordering the slices in the pie chart by size; when the bar chart is used the comparison is much easier.. Similarly, comparisons between data sets are easier using the bar chart. However, if the goal is to compare a given category (a slice of the pie) with the total (the whole pie) in a single chart and the multiple is close to 25 or 50 percent, then a pie chart can often be more effective than a bar graph.
Variants and similar charts
Polar area diagram
The polar area diagram is similar to a usual pie chart, except sectors are equal angles and differ rather in how far each sector extends from the center of the circle. The polar area diagram is used to plot cyclic phenomena (e.g., count of deaths by month). For example, if the count of deaths in each month for a year are to be plotted then there will be 12 sectors (one per month) all with the same angle of 30 degrees each. The radius of each sector would be proportional to the square root of the death count for the month, so the area of a sector represents the number of deaths in a month. If the death count in each month is subdivided by cause of death, it is possible to make multiple comparisons on one diagram, as is clearly seen in the form of polar area diagram famously developed by Florence Nightingale.
The first known use of polar area diagrams was by AndrÃ©Michel Guerry, which he called courbes circulaires, in an 1829 paper showing seasonal and daily variation in wind direction over the year and births and deaths by hour of the day. LÃ©on Lalanne later used a polar diagram to show the frequency of wind directions around compass points in 1843. The wind rose is still used by meteorologists. Nightingale published her rose diagram in 1858. The name "coxcomb" is sometimes used erroneously. This was the name Nightingale used to refer to a book containing the diagrams rather than the diagrams themselves. It has been suggested that most of Nightingale's early reputation was built on her ability to give clear and concise presentations of data.
Spie chart
A useful variant of the polar area chart is the spie chart designed by Feitelson . This superimposes a normal pie chart with a modified polar area chart to permit the comparison of a set of data at two different states. For the first state, for example time 1, a normal pie chart is drawn. For the second state, the angles of the slices are the same as in the original pie chart, and the radii vary according to the change in the value of each variable. In addition to comparing a partition at two times (e.g. this year's budget distribution with last year's budget distribution), this is useful for visualizing hazards for population groups (e.g. the distribution of age and gener groups among road casualties compared with these groups's sizes in the general population). The R Graph Gallery provides an example.
Multilevel pie chart
Multilevel pie chart, also known as a radial tree c
Connected Mathematics is a comprehensive, problemcentered curriculum designed for all students in grades 68 based on the NCTM standards. The curriculum was developed by the [http://connectedmath.msu.edu/Connected Mathematics Project (CMP)] at Michigan State University and funded by the National Science Foundation.
Each grade level curriculum is a fullyear program, and in each of the three grade levels, topics of number, algebra, geometry/measurement, probability and statistics are covered in an increasingly sophisticated manner. The program seeks to make connections within mathematics, between mathematics and other subject areas, and to the real world. The curriculum is divided into units, each of which contains investigations with major problems that the teacher and students explore in class. Extensive problem sets are included for each investigation to help students practice, apply, connect, and extend these understandings.
Connected Mathematics addresses both the content and the process standards of the NCTM. The process standards are: Problem Solving, Reasoning and Proof, Communication, Connections and Representation. For example, in Moving Straight Ahead students construct and interpret concrete, symbolic, graphic, verbal and algorithmic models of quantitative and algebraic relationships, translating information from one model to another.
Like other curricula implementing the NCTM standards, Connected Math has been criticized by supporters of traditional mathematics for not directly teaching standard arithmetic methods.
Research Studies
One 2003 study compared the mathematics achievement of eighth graders in the first three school districts in Missouri to adopt NSFfunded Standardsbased middle grades mathematics curriculum materials (MATH Thematics or Connected Mathematics Project) with students who had similar prior mathematics achievement and family income levels from other districts. Significant differences in achievement were identified between students using Standardsbased curriculum materials for at least 2 years and students from comparison districts using other curriculum materials. All of the significant differences reflected higher achievement of students using Standardsbased materials. Students in each of the three districts using Standardsbased materials scored higher in two content areas (data analysis and algebra), and these differences were significant.
Another study compared statewide standardized test scores of fourthgrade students using Everyday Mathematics and eighthgrade students using Connected Mathematics to test scores of demographically similar students using a mix of traditional curricula. Results indicate that students in schools using either of these standardsbased programs as their primary mathematics curriculum performed significantly better on the 1999 statewide mathematics test than did students in traditional programs attending matched comparison schools. With minor exceptions, differences in favor of the standardsbased programs remained consistent across mathematical strands, question types, and student subpopulations.
Controversy
As one of many widely adopted curricula developed around the NCTM standards, Connected Mathematics has been criticized by advocates of traditional mathematics as being particularly ineffective and incomplete and praised by various researchers who have noted its benefits in promoting deep understanding of mathematical concepts among students. In a review by critic James Milgram, "the program seems to be very incomplete... it is aimed at underachieving students." He observes that "the students should entirely construct their own knowledge.. standard algorithms are never introduced, not even for adding, subtracting, multiplying and dividing fractions." However, studies have shown that students who have used the curriculum have "develop[ed] sophisticated ways of comparing and analyzing data sets, . . . refine[d] problemsolving skills and the ability to distinguish between reasonable and unreasonable solutions to problems involving fractions, . . . exhibit[ed] a deep understanding of how to generalize functions symbolically from patterns of data, . . . [and] exhibited a strong understanding of algebraic concepts and procedures," among other benefits.
Districts in states such as Texas were awarded NSF grants for teacher training to support curricula such as CM. Austin ISD received a $5 million NSF grant for teacher training in 1997. NSF awarded $10 million for "Rural Systemic Initiatives" through West Texas A&M. At the state level, the SSI (Statewide Systemic Initiative), was a federallyfunded program developed by the Dana Center at the University of Texas. Its most important work was directing the implementation of CM in schools across the state. But in 1999, Connected Mathematics was rejected by California's revised standards because it was judged at least two years below grade level and it contained numerous errors . After the 20002001 academic year, state monies can no longer be used to buy Connected Mathematics
The Christian Science Monitor noted parents in Plano Texas who demanded that their schools drop use of CM, while the New York Times reported parents there rebelled against folding fraction strips rather than using common denominators to add fractions. For the improved second edition, it is stated that "Students should be able to add two fractions quickly by finding a common denominator". The letter to parents states that students are also expected to multiply and divide fractions by standard methods.
What parents often do not understand is that students begin with exploratory methods in order to gain a solid conceptual understanding, but finish by learning the standard procedures, sometimes by discovering them under teacher guidance. Largescale studies of reform curricula such as Connected Mathematics have shown that students in such programs learn procedural skills to the same level as those in traditional programs, as measured by traditional standardized tests. Students in standardsbased programs gain conceptual understanding and problemsolving skills at a higher level than those in traditional programs.
Despite disbelief on the part of parents whose textbooks always contained instruction in mathematical methods, it is claimed that the pedagogical benefits of this approach find strong support in the research: "Over the past three to four decades, a growing body of knowledge from the cognitive sciences has supported the notion that students develop their own understanding from their experiences with mathematics."
Examples of criticism
Connected Mathematics treatment of some topics include exercises which some have criticized as being either "subjective" or "having nothing to do with the mathematical concept" or "omit standard methods such as the" formula for arithmetic mean. (See above for discussion of reasons for initial suppression of formulas.) The following examples are from the student textbooks, which is all the parents see. (See discussion below.)
Average
In the first edition, one booklet focuses on a conceptual understanding of median and mean, using manipulatives. The standard algorithm was not presented. Later editions included the algorithm.
Comparing fractions
In the 6th grade u
In chemistry, mole fractionx is a way of expressing the composition of a mixture. The mole fraction of each component i is defined as its amount of substancen_{i}divided by the total amount of substance in thesystem, n
 x_i \ \stackrel{\mathrm{def}}{=}\ \frac{n_i}{n}
where
 n = \sum_i n_i \,
The sum is over all components, including the solvent in the case of a chemical solution. As an example, if a mixture is obtained by dissolving 10 moles of sucrose in 90 moles of water, the mole fraction of sucrose in that mixture is 0.1.
The same value for the mole fraction ratio is obtained using the number of molecules of i, N_{i}, and the total number of molecules of all kinds, N, since
 N_i = n_i\, N_{\rm A}\,
where N_{A} is the Avogadro constantâ‰ˆ 6.022 mol. By definition, the sum of the mole fractions equals one, a normalization property.
 \sum_i x_i \ \stackrel{\mathrm{def}}{=}\ 1 \,
Simple representation
The simple representation of examining the mole fraction is thinking in terms of A and B. The mole fraction of A would be moles of A divided by the moles of A and moles of B. This way, adding the two mole fractions together would equal one.
 \mathrm{Mole\ fraction\ of\ A} =\frac{\mathrm{Moles\ of\ A}}{\mathrm{Moles\ of\ A\ +\ Moles\ of\ B}}
Another representation would be:
 \mathrm{Mole\ fraction\ of\ A}=\frac{nA}{nA + nB}
Where nA is the number of moles of substance A and nB is the number of moles of substance B
Notes and qualifications
Mole fractions are dimensionless numbers. Other ways of representing concentrations, e.g., molarity and molality, yield dimensional quantities (per litre, per kilogram, etc.). When chemical formulas seem to be taking the logarithms of dimensional quantities, there is an implied ratio, and such expressions can always be rearranged so that the arguments of the logarithms are dimensionless numbers, as they must be.
Mole fractions are one way of representing the concentrations of the various chemical species. They are an idealmixture approximation to the effect of concentration on the equilibrium or rate of a reaction. In practice (except for very dilute solutions or for gases at atmospheric pressure), all measures of concentration must be multiplied by correction factors called activity coefficients in order to yield accurate results.
The mole fraction is sometimes denoted by the lower case Greek letter Ï‡ (chi) instead of aRomanx. For mixtures of gases, it is more usual to use the letter y. The mole fraction of a substance in a reaction is also equal to the partial pressure of that substance.
Mole fraction is based upon moles of substances, not moles of ions. For example, 4 moles of NaCl would not be dissociated into 4 moles of sodium ions and 4 moles of chloride ions.
In mathematics, the term global field refers to either of the following:
 an algebraic number field, i.e., a finite extension of Q, or
 a global function field, i.e., the function field of an algebraic curve over a finite field, equivalently, a finite extension of F_{q}(T), the field of rational functions in one variable over the finite field with q elements.
An axiomatic characterization of these fields via valuation theory was given by Emil Artin and George Whaples in the 1940s.
There are a number of formal similarities between the two kinds of fields. A field of either type has the property that all of its completions are locally compact fields (see local fields). Every field of either type can be realized as the field of fractions of a Dedekind domain in which every nonzero ideal is of finite index. In each case, one has the product formula for nonzero elements x:
 \prod_v x_v = 1.\
The analogy between the two kinds of fields has been a strong motivating force in algebraic number theory. The idea of an analogy between number fields and Riemann surfaces goes back to Richard Dedekind and Heinrich M. Weber in the nineteenth century. The more strict analogy expressed by the 'global field' idea, in which a Riemann surface's aspect as algebraic curve is mapped to curves defined over a finite field, was built up during the 1930s, culminating in the Riemann hypothesis for local zetafunctions settled by AndrÃ© Weil in 1940. The terminology may be due to Weil, who wrote his Basic Number Theory (1967) in part to work out the parallelism.
It is usually easier to work in the function field case and then try to develop parallel techniques on the number field side. The development of Arakelov theory and its exploitation by Gerd Faltings in his proof of the Mordell conjecture is a dramatic example.
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Answers:1. spent = operation minus, or multiply by given fraction. so, for remaining money, subtract from starting amount, or multiply by 1fraction e.g., start with X spent 1/3 of it, so he has 2/3 * X remaining. spent 3/4 of that, so he has 1/4 remaining, i.e., 1/4 * 2/3 * X. spent 20, so has amount  20 left. 2. Julia saw a dress for 20% off. The cashier took off an additional $25 at the checkout. The final price was $35. What was the original price? 3. Racer X is 25 years older than Speed Racer. Trudy is 4/5 of Speed Racer's age. If Trudy is 35 years old, write an expression for how old Racer X is.
Answers:15  34x + 8 = (x  2)(15x  4)
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