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Primary mathematics/Fractions

In the early primary grades, fractions are generally thought of as "a part" of a whole, and are used to express numbers less that one. Yet, fractions encompass much more mathematical territory than this. They can of course be larger than one.

Methodology
Generally speaking, the progression used when teaching fractions starts with the use of manipulatives and other visual representations that make connections to real life things: pizza, money, candy, etc. As children begin to be able to articulate these understandings both verbally and with mathematical expressions, more abstract models such as area rectangles, fraction bars and grids are introduced, permitting students to expand their understandings in more abstract ways.

The Use of Manipulatives to Teach Fractions
Candidates for manipulatives used to teach fractions are single objects or groups of the same object that can be divided equally. There are a number of commercially available "toys" that are designed just for this purpose, but these are not necessary for a resourceful teacher. Regardless, students should initially be making connections to things they "know", like pizza and candy.

Origami
By simply taking a square piece of paper and folding it in half; write the fraction $$\tfrac {1}{2}$$; fold in half again; write $$\tfrac {1}{4}$$. Continue this process to create smaller fractions. Unfold the paper and trace lines where there are folds in the paper.

Notice that in the case of the example, the manipulative (the folding paper) becomes a model (a more abstract visual representation) as students draw over the folds and fill in the values.

Money
As we all know, money is a great motivator (for any age). It is something all children are familiar and comfortable with, and perhaps most importantly - they want it! This doesn't mean you have to give it to them, but you can use it to leverage their interest.

The use of money as a manipulative/model to teach fractions (and many other concepts in mathematics) has many advantages. Thankfully, most all currencies use a base ten system. Using change, the US dollar can be broken up into fourths (quarters), tenths, twentieths and hundredths, which makes it a perfect model to use for teaching decimals and even percentages as well.

From Manipulatives to Models
While manipulatives offer a tactile and visual way for students to learn about fractions, models let them take their exploration to a higher level of abstraction, while still maintaining a visual aspect to their learning.

Area Diagrams
"Area diagrams" or "area rectangles" are perhaps one of the best ways for children to model fractions. They have the advantage of visually showing "how much" of the whole a fraction takes up, and will become an indespensible tool students will eventually use for modeling operations with fractions. Students should be encouraged to create their own area diagrams to reinforce their learning.

"100 grids" are a type of area diagram commonly used to teach fractions. Because the fractions that 100 grids model at the single unit level are hundreths, they are usually introduced in the upper primary grades. Similar to the way that area rectangles can be used to model multiplication, 100 grids are a great way to model fractions and simultaneously facilitate students in making important connections to percentages and decimals. Be sure to have students demonstrate that they understand these connections by filling in the 100 grid and then writing down the equivalent expressions.

For instance: $$\tfrac{1}{4}=25\%=0.25$$

Notice in the above example that children should be expected to always reduce fractions to simplest terms (if they know how).

Equivalent Fractions
Because fractions represent both division and ratios, it is important to introduce all of these concepts to their web, or schema, of mathematical understanding.

Equivalent Fractions and Ratios
When students create tables such as the ones below (based on the multiplication tables of the numbers 1 and 3, and the cost of a candy bar) they are exposed to the connection between fractions and ratios.

In the table below, note that after division is used to "reduce" the ratio to ascertain the cost of a single candy bar, the rest of the table is much more easily completed:

Using the Identity Property of Multiplicaton to Create Equivalent Fractions
Students need to understand the reason that the methods they use to create equivalent fractions actually work. One very effective way to do this is for them to initially learn the Identity Property of Multiplication. Of course, we know that $$\tfrac{a}{a}=1$$, but most primary grade students are not able to make the connection between this axiom in action and its corresponding abstract mathematical expression stated with variables, so we should first let children explore the ways they can represent the number $$\mathbf{1}$$: I.e., $$\tfrac{23}{23}=1$$, or   $$\tfrac{0.6}{0.6}=1$$,  or   $$\tfrac{1,000,000}{1,000,000}=1$$... We can explain to students that if $$\frac{7}{7}=1$$, and anything multiplied by one doesn't change its value, then $$\frac{3}{5}\times\frac{7}{7}=$$ shouldn't change it's value, but this is very difficult for many students to understand because of the level of abstraction. Models and visual representations become necessary. One very effective visual representation we can use is the Giant One.

Students also need to be reminded of the fact that anything times 1 doesn't change its value (The Identity Property of Multiplication).

The Giant One can be used to set up the equation, which helps students to visualize a way to find equivalent fractions. Students should draw these themselves to reinforce their understanding of the concept. Even students with poor motor skills can draw Giant Ones; it doesn't matter how neatly they are drawn:

Using Area Rectangles to Create Equivalent Fractions
As we have seen earlier, area rectangles have the advantage of visually showing students "how much" a fraction represents. And because it is possible to show a fraction using either horizontal or vertical lines, by using both horizontal and vertical lines it is possible to simultaneously show two superimposed fractions.

Note how this exercise is executed almost in the same fashion as multiplying fractions, except that the second fraction, because it is equivalent to 1 is not filled in. The super-imposition of thirds and forths creates a total of 12 equal rectangles, or 12ths, and 9 filled in rectangles. Students should become adept at creating these. Students eventually should be able to look at the image on the right and know what equivalent fractions are being modeled.

The same thing should be modeled with a Giant One problem to reinforce the connection:

Basic Operations
The more standard algorithms that most of us use to add, subtract, and multiply fractions, are to a child's perspective, a collection of rules, which at best "work", but at worst become all mixed up in students' minds. This is usually because these 'rules' have no underpinnings in the students' web of mathematical knowledge. In other words, if students don't understand why the "rules" for using these methods exist in the first place, it becomes very difficult to apply them at the right time, not to mention execute them consistently. For this reason, this "traditional" approach (just teaching them how to do things, but not why they are doing them the way they are) to teaching operations with fractions requires a "drill and kill" methodology.

Thankfully, the contemporary approach to teaching these methods has students explore these concepts in many different ways at a fundamental level.

Adding Fractions
In the lowest grades, it may be best to only add fractions with a common denominator. This can be explained as "you can't add apples and oranges together", or, since those could be added together to get the total pieces of fruit, you could use an example of adding items which are even more different, like weeks and sheets of paper. For example, "if you have 2 weeks and 10 pieces of paper to make an art project, you don't add them together to get 12". In this analogy, the denominator of the fractions describes the type of fractions they are, which determines if they can be added.

In later grades, students can convert one or both fractions to have a common denominator, before adding. This should wait until after multiplication of fractions has been taught, as this is a necessary step in finding a common denominator.

Subtracting Fractions
The same logic applies to teaching subtraction of fractions as addition, with one additional element; it is now possible to get a negative result. This can be explained with real world objects and "owing". For example, "if you said you would give your brother half a cookie, but only had a quarter cookie to give him, you still owe him another quarter cookie. This is the same as saying you now have a negative quarter cookie".

Multiplying Fractions
Most students when first confronted with the problem $$\tfrac{2}{3}\times\tfrac{1}{4}$$ have difficulty understanding that the product of these numbers will be smaller than either of them. This is a natural conjecture, born of their years (not too many) of experience with multiplication.

The use of the word "of" to describe multiplication needs to be used to realign their understanding. Children know that $$5\times6=30$$ because five groups of six is thirty. Asking them "what $$\tfrac{1}{4}$$ of 100 is" helps them to better understand why multiplying with fractions changes their earlier understanding. As they begin to understand the implications of this they can then be introduced to the concept using two fractions. I.e., what is 1/2 of 1/2? At this point models should be used to reinforce their understanding of this. Area models can be very effective towards this end, where the area of the intersecting fractions represents the product.

Students should be both asked to create these models from word problems as well as create word problems that fit with the model. For instance, 2/3 of the students in the class are wearing yellow shirts. 1/4 of the class goes to the library. What fraction of the class is wearing yellow shirts in the library? or 2/3 of my garden was planted with yellow flowers, and 1/4 of all of the flowers were sold. What fraction of all of the flowers sold from the garden were probably yellow? Note that in this case the wording of the problem has created a lesson with a connection to probability.

The image below models the problem $$\frac{4}{10}\times\frac{5}{10}$$ using a 100 grid.

Note that the use of a 100 grid permits the answers to problems to be expressed as fractions, decimals and even percentages (E.g., $$.4\times.5$$ and 40% of 1/2.)

Dividing Fractions
Teaching students how to divide with fractions by simply showing them how to multiply the dividend and the reciprocal of the divisor offers practically no insight to their mathematical understanding.. because no connections have been made to what they already know about division. Before students learn to divide whole numbers, they need to understand how to multiply them. The same concept applies with fractions. Students should first explore division of whole numbers by fractions at a basic level. Models should be used whenever possible. For instance, the problem "How many 1/2s are there in 4? can be modeled with any graphic representation, such as.

The next level of difficulty involves two numbers with fractions, such as can be found in the problem, "How many $$\tfrac{2}{3}$$ are there in $$2\tfrac{2}{3}$$?

Similar models can be used to explore division of fractions with different denominators. I.e., "How many $$\tfrac{1}{6}$$ are there in $$2\tfrac{2}{3}$$?

At this time students should also be exploring and learning how to find reciprocals. It will be necessary for them to understand what reciprocals do before they start to use them to accomplish division problems with fractions.

Rulers
The above method of teaching division with fractions can be accomplished with rulers as well. Most rulers that are incremented in inches provide fractions down to sixteenths. Rulers are a "real world" tool and familiarity with the specific fractions encountered when using a ruler have an exponential component. I.e, $$\frac{1}{2}$$ $$\frac{1}{2^2}$$  $$\frac{1}{2^3}$$  $$\frac{1}{2^4}$$

Division with Giant One
Though to a teacher, it may at first seem complex, students who are comfortable with the Giant One find this method is useful if not preferable when first solving division problems that include fractions. The best way to teach this method is to start with a whole number dividend and express the problem as a fraction(which students understand is the same as division). Thus, the equation $$4\div\tfrac{2}{3}$$, turns into the fraction $$\frac{4}{\tfrac{2}{3}}$$. Multiplying this fraction by a Giant One such that the number of the divisor of the bottom fraction is inserted into the Giant One creates a model like the one at right, which becomes $$12\div2$$.

Division with two fractions requires the use of the reciprical of the divisor of the problem to be inserted into the Giant One:

Eventually, students who use this method recognize that the top fracton of the problem is being multiplied by the reciprical of the bottom fraction. The process of discovering this "trick" for themselves gives them ownership of the larger conceptual understanding. In other words, students who learn to divide fractions in this way understand at a fundamental level "why" the trick works

Adding fractions
Fractions that have the same or "Common" denominator are called "Like" fractions.

1/3, 2/3

(one-third, two-thirds)

To add Like fractions together such as these:

1/3 + 2/3 = ?

(one-third plus two-thirds equals what?)

1. Add the numerators (the top numbers):

1 + 2 = 3

(one plus two equals three)

2. Use the common denominator (the bottom numbers):

/3

All together it looks like this:

1/3 + 2/3 = 3/3

(one-third plus two-thirds equals three-thirds)

3. Then simplify the answer as much as you can by dividing the denominator into the numerator.

3/3 = 1

(three divided by three equals one)

If the numerator is now larger than the denominator it might look like this:

4/5 + 3/5 = 7/5

(four-fifths plus three-fifths equals seven-fifths)

simplified:

5 goes into 7 one time, with 2 left over. Put the remainder (2) over the denominator (5).

7/5 = 1 2/5

(seven-fifths equals one and two-fifths)

To Add or Subtract fractions with different denominators, you must first convert them to equivalent fractions with common denominators.

title: Primary mathematics/Fractions


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Primary mathematics/Working with fractions

Mixed numbers and top heavy fractions:

A mixed number is an integer and a fraction, eg 2 and a half, a top heavy fraction is a fraction with a larger numerator than demominator, eg 3/2. For multiplication and division you must not use mixed numbers. They can be used in addition and subtraction (do the integers and fractions seperately and combine the result) if this is not too confusing.

Simplest terms: Multiplying (or dividing) the numerator and denominator by the same number will not change the fraction. A fraction is put into its simplest terms by dividing both by the largest integer that goes into both a whole number of times, this can be done in stages using small integers such as 2 and 3, 5 and 10 repeatedly.

Multiplication:

$$\frac{a}{c}*\frac{b}{d} = \frac{ab}{cd}$$

Where ab=a*b.

Division:

To divide fractions; remember that a fraction is a division. Fractions can be stacked like so:

$$\frac{\frac{a}{b}}{\frac{c}{d}}$$

The rule to remember to simplify is that anything under an odd number of lines goes on the bottom and anything under an even number of lines goes on the top (0 is even for this purpose) thus:

$$\frac{ad}{bc}$$

It is easier (perhaps) to remember that the second fraction in the division gets turned upside down.

Addition and subtraction:

For addition the denomintators of the two fractions must be the same - if this is so then simply add (or subtract) the numerators:

$$\frac{a}{b}+\frac{c}{b} = \frac{a+c}{b}$$

If the denominators are not the same then a common denominator must be found, the simplest way to do this is take the product of the denominators allthough if one is a multiple of the other simply multiplying it by the aplicable integer works aswell. Remember to multiply the numerators by the same number. In general:

$$\frac{a}{b}+\frac{c}{d} = \frac{ad+cb}{bd}$$

But for example

$$\frac{a}{5}+\frac{b}{10} = \frac{2a+b}{10}$$

Is possible.

Remember that young children will not understand the algebraic formulation here. The art of primary teaching is finding a way to make this make sense to young children however teachers need to know these general rules.

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/


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Primary mathematics/Powers, roots, and exponents

Exponents
Exponents, or powers, are a way of indicating that a quantity is to be multiplied by itself some number of times. In the the expression 25, 2 is called the base and 5 is called the exponent, or power. 25 is shorthand for "multiply five twos together": 25 = 2×2×2×2×2 = 32. Notice that the exponent tells us how many bases to multiply, not how many multiplications to perform. (In fact, the number of multiplications is one less than the number of bases.) 25 is read "two raised to the fifth power" or simply "two to the fifth."

In general,$$x^n = \underbrace{x \cdot x \cdot x \cdot \,\,\cdots\,\, \cdot x}_{n factors}$$ where there are n x's to be multiplied.

It is better to say that exponent is 1 multiplied by x n times, $$x^n = 1 \cdot \underbrace{x \cdot x \cdot x \cdot \,\,\cdots\,\, \cdot x}_{n factors}$$. This is permissible because factor 1 is multiplicative identity, adding it does not change the product yet enables the  empty one, $$x^0 = 1$$.

The definition of exponent corresponds to equivalent definition of multiplication: $$x\times$$ n is nothing more than x added to 0 n times, $$x \cdot n = 0 + \underbrace{x + x + \cdots + x}_{n factors}.$$ You can even convert one to another, using, $$x ^ n= x \cdot x \cdot \ldots \cdot x = a^\chi \cdot a^\chi \cdot \cdots \cdot a^\chi = a^{\chi +\chi +\cdots +\chi} = a^{\chi \cdot n}.$$

Squaring numbers
A convenient way to say that a number is to be multiplied by itself (for example 5×5) is to say that the number is squared. To help visualize this, picture a square which is 5 units long and 5 units wide. This square will then have an area of 52 or 25 square units. 52 can be read "five raised to the second" but is most commonly read "five squared."

Area of five rows by five columns = 5 × 5 = 52 = 25.

Table of perfect squares
You may want to memorize some perfect square numbers:

Cubing numbers
Similarly, a convenient way to say that a number is to be multiplied by itself and then itself again (for example 5×5×5) is to say that the number is cubed. To help visualize this, picture a cube which is 5 units long, 5 units wide, and 5 units high. This cube will then have a volume of 53 or 125 cubic units. 53 can be read "five to the third power" but is commonly read "five cubed."

Volume of a cube with height, width, and length of five = 5 × 5 × 5 = 53 = 125.

Table of perfect cube numbers
You may want to memorize some perfect cube numbers:

Higher powers
Numbers higher than three may also be used as exponents, although there is no common term for numbers raised to a fourth power or higher. For example, 54 = 5×5×5×5 = 625.

Table of higher powers
Note: 00 is undefined.

Properties of Exponents
There are several properties of exponents which are frequently used to manipulate and simplify algebraic and arithmetic expressions.

It follows immediately that

and

These can be extended to multiple factors

$$ \begin{align}

x^{m_1}x^{m_2}x^{m_3} \cdots x^{m_n} & = x^{m_1+m_2+m_3+ \, \cdots \, +m_n}\\

(x_1x_2x_3 \cdots x_n)^m & = x_1^mx_2^mx_3^m \cdots x_n^m\\

( \cdots (((x^{m_1})^{m_2})^{m_3}) \cdots )^{m_n} & = x^{m_1m_2m_3 \cdots \, m_n}

\end{align} $$

Alternatively, we can use first property to obtain

and use others to write down

Bases and exponents of one and zero
Any number raised to an exponent of one equals itself. So, for example, 51 = 5.

Any non-zero number raised to an exponent of zero equals one. So, for example, 50 = 1.

Zero raised to any positive exponent is still zero. So, for example, 05 = 0.

One raised to any exponent is still one. So, for example, 15 = 1.

Zero raised to an exponent of zero is not defined.

Fractional powers
A base may also be raised to a fractional, or decimal power. This will be covered later in the lesson.

Roots
Roots are the inverse operation of powers (i.e. $$\sqrt[n]{x^n} = x$$ and from the properties of exponents we have $$\sqrt[n]{x} = x^{1/n}$$):

Square root
The inverse operation of squaring a number is taking the square root of that number. So, for example, the square root of 25 is the number which must be multiplied by itself to equal 25. In this case, the answer is 5. There are two types of notation used here:

$$\sqrt{25} = 25^{1/2} = 5$$

Note, however, that most square roots don't yield integers, and many don't even produce rational numbers.

Manually finding a square root
One method for manually taking square roots is to repeatedly do long division. Let's take the square root of 10 in this example. We would start by estimating the answer. Since 32 = 9 and 42 = 16, we know the answer lies between 3 and 4. Furthermore, since 10 is only one away from 9, but is 6 away from 16, we could estimate that the answer is one-seventh of the way between 3 and 4. This won't give an exact answer, and a seventh is ugly to work with, so let's use a fifth, instead. This gives us 3 1/5 or 3.2 as a starting estimate.

Now do long division to divide 10 by 3.2. We get 3.125. The average of 3.2 and 3.125 is (3.2 + 3.125)/2 = 6.325/2 = 3.1625, so that will be our next estimate.

Now do long division to divide 10 by 3.1625. We get 3.162055... (we didn't really need to go more than one digit beyond the number of decimal places we started with). The average of 3.1625 and 3.1621 is 3.1623, so that will be our next estimate.

Now do long division to divide 10 by 3.1623. We get 3.162255...

So, this method can be repeated to get the desired level of accuracy. The actual square root of 10 is 3.16227766...

Note that calculators or computers are used for most square root calculations, but knowing how to manually calculate a square root can be quite useful when no calculator is available.

If you would like to try this method yourself, try finding the square root of 7.

Cube root
The inverse operation of cubing a number is taking the cube root of that number. So, for example, the cube root of 125 is the number which must be multiplied by itself and then multiplied by itself again to equal 125. In this case, the answer is 5. There are two types of notation used here:

$$\sqrt[3]{125} = 125^{1/3} = 5$$

Note, however, that most cube roots don't yield integers, and many don't even produce rational numbers.

Higher roots
Numbers higher than three may also be used as roots, although there is no common term for fourth roots or higher. For example:

$$\sqrt[4]{625} = 625^{1/4} = 5$$

Note, however, that most higher roots don't yield integers, and many don't even produce rational numbers.

Combining powers and roots
The unit fraction notation used for roots previously may have given you the idea that roots are really the same as powers, only with a unit fraction (one over some number) instead of an integer as the exponent. Thus, the fractional notation is actually preferred in higher mathematics, although the root symbol is still used occasionally, especially for square roots.

Fractions as exponents
Other (non-unit) fractions may also be used as exponents. In this case, the base number may be raised to the power of the numerator (top number in the fraction) then the denominator (bottom number) may be used to take the root. For example:

$$8^{2/3} = \sqrt[3]{8^2} = \sqrt[3]{64} = 4$$

Alternatively, you can take the root first and then apply the power:

$$8^{2/3} = (\sqrt[3]{8})^2 = 2^2 = 4$$

Decimal exponents
Any fractional exponent can also be expressed as a decimal exponent. For example, a square root may also be written as:

$$25^{1/2} = 25^{0.5} = 5 \;$$

Also, decimals which can't be expressed as a fraction (irrational numbers) may be used as exponents:

$$5^{3.1415926...} = 156.99...\;$$

Such problems aren't easy to solve by hand using basic math skills, but the answer can be estimated manually. In this case, since 3.1415926 is between 3 and 4 (and considerable closer to 3), we know that the answer will be between 5^3 (or 125) and 5^4 (or 625), and considerable closer to 125.

Negative exponents
A negative exponent simply means you take the reciprocal (one over the number) of the base first, then apply the exponent:

$$25^{-0.5} = 1/25^{0.5} = 1/5 \;$$

Alternatively, you can first apply the exponent (ignoring the sign), then take the reciprocal:

$$25^{-0.5} = 5^{-1} = 1/5 \;$$

Fractions as bases
When a fraction is raised to an exponent, both the numerator and denominator are raised to that exponent:

$$(5/6)^2 = 5^2/6^2 = 25/36 \;$$

Fractions may also be used for both the base and exponent:

$$(25/36)^{1/2} = 25^{1/2}/36^{1/2} = 5/6 \;$$

In addition, negative fractional exponents may be used, taking the reciprocal of the base, as always, to find the solution.

Negative bases
Negative bases can be handled normally for integer powers:

$$(-5)^2 = (-5) \times (-5) = 25 \;$$

$$(-5)^3 = (-5) \times (-5) \times (-5) = -125 \;$$

$$(-5)^4 = (-5) \times (-5) \times (-5) \times (-5) = 625 \;$$

$$(-5)^5 = (-5) \times (-5) \times (-5) \times (-5) \times (-5) = -3125 \;$$

Note that negative bases raised to even powers produce positive results, while negative bases raised to odd powers produce negative results.

Be careful with negative signs. Since -5 = -1×5, there is a difference between $$ -5^2$$ and $$(-5)^2$$. The former means the negative of 5 times 5, whereas the latter means -5 squared. In other words,

$$-5^2 = -1\times5^2 = -1\times(5\times5) = -1\times25 = -25$$

but

$$(-5)^2=(-5)\times(-5)=25$$

Roots and fractional/decimal powers are a bit trickier. Odd roots work out fine:

$$(-125)^{1/3} = -5 \;$$

Even roots, however, have no real solution:

$$(-25)^{1/2} = ? \;$$

$$(-625)^{1/4} = ? \;$$

Note that there is no real number, when multiplied by itself, which will produce -25, because 5×5 = 25 and -5×-5 = 25. There is actually a solution, called an imaginary number, but that won't be discussed until later lessons.

Principal root
Note that, since both 5×5 = 25 and -5×-5 = 25, when we are asked to take the square root of 25 there are, in fact, two valid answers, 5 and -5. Actually, any even root of a positive number will have two solutions, one positive and one negative. This may seem unusual, but, in higher mathematics, problems often have multiple solutions.

However, for many problems, only the positive value seems to physically work. For example, if we are asked to figure the length of the sides of a square yard which has an area of 25 square units, only 5 units on a side works. If we said "each side can also have a length of -5 units", that doesn't make any sense. For this reason, the positive solution is called the principal root, and, depending on the question, may be the only desired answer. In cases where either answer is valid, it is sometimes written as ±5 (read as "plus or minus five").


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Primary mathematics/Average, median, and mode

Introduction
There are three primary basic measures of central tendency, and a couple less often used measures, which each, in their own way, tell us what a typical value is for a set of data. After these, some more advanced measures and notation will be introduced.

Mode
The mode is simply the number which occurs most often in a set of numbers. For example, if there are ten 12-year-olds in a class, seven 13-year-olds, and four 14-year-olds, the mode is 12, since there are more 12-year-olds than any other age.

Average
The straight average, or arithmetic mean, is the sum of all values divided by the number of values. For example, if Natasha scored 91%, 85%, 99% and 77% on a test, the average is (91%+85+99+77)=352 Then divide to arrive at the average for the 4 test. 352/4=88. 88 is the average of the 4 test Natasha took.

Weighted average
The weighted average or weighted mean, is similar to the straight average, with one exception. When totaling the individual values, each is multiplied by a weighting factor, and the total is then divided by the sum of all the weighting factors. These weighting factors allow us to count some values as "more important" in finding the final value than others.

Example
Let's say we had two school classes, one with 20 students, and one with 30 students. The grades in each class on a particular test were:

 * Morning class = 62, 67, 71, 74, 76, 77, 78, 79, 79, 80, 80, 81, 81, 82, 83, 84, 86, 89, 93, 98

 * Afternoon class = 81, 82, 83, 84, 85, 86, 87, 87, 88, 88, 89, 89, 89, 90, 90, 90, 90, 91, 91, 91, 92, 92, 93, 93, 94, 95, 96, 97, 98, 99

The straight average for the morning class is 80% and the straight average of the afternoon class is 90%. If we were to find a straight average of 80% and 90%, we would get 85% for the mean of the two class averages. However, this is not the average of all the students' grades. To find that, you would need to total all the grades and divide by the total number of students:

\frac{4300\%}{50} = 86\% $$

Or, you could find the weighted average of the two class means already calculated, using the number of students in each class as the weighting factor:

\frac{(20)80\% + (30)90\%}{20 + 30} = 86\% $$

Note that if we no longer had the individual students' grades, but only had the class averages and the number of students in each class, we could still find the mean of all the students grades, in this way, by finding the weighted mean of the two class averages.

Geometric mean
The geometric mean of two numbers is a number midway between two values by multiplication, rather than by addition. For example, the geometric mean of 3 and 12 is 6, because you multiply 3 by the same value (2, in this case) to get 6 as you must multiply by 6 to get 12. The mathematical formula for finding the geometric mean of two values is:

$$\sqrt{AB}$$

Where:

 * A = one value
 * B = the other value

So, in our case:

$$\sqrt{3(12)} = \sqrt{36} = 6$$

Note the new notation used to show multiplication. We now can omit the multiplication sign and show simply AB to mean A×B. However, when using numbers, 312 would be confusing, so we put parenthesis around at least one of the numbers to make it clear.

Also, notice that the geometric mean can only be found between two values, not using three or more.

Mode
In elections, the mode is often called the plurality, and the candidate who gets the most votes wins, even if they don't get the majority (over half) of the votes.

Median
Medians tend to be used with unevenly distributed data, such as incomes. Let's say we had ten people, with the following incomes:

$  1000 $   2000 $   3000 $   4000 $   5000 $   6000 $   7000 $   8000 $   9000 $1000000  $1045000 = Total

The arithmetic mean (straight average) would be $1,045,000/10 or $104,500, giving the impression that, on average, they have high incomes, when only one does. The median would be ($5000+$6000)/2, or $5500, and that more accurately reflects the typical income.

Average
The straight average, or arithmetic mean, is perhaps the most often used way to find a "typical value" of a set of data.

Weighted average
As previously noted, the weighted average is used to weigh some data points more heavily than others. For instance, if quizzes make up 20%, tests 35%, and a term paper 45% of a final grade, we cannot simply take the arithmetic mean. Also note that since the quizzes make up 20% of the grade, in contrast to the other subjects, it is of least importance when considering all three of them. So, each grade is weighted differently: firstly, take the arithmetic mean of each constituent for both semesters. Suppose the following table makes up our grades in terms of each constituent: To take the arithmetic mean of each constituent, add each subject's grade in both semesters and divide by 2. The arithmetic mean of the quizzes would be:

$$\frac{82+88}{2}$$

82 + 88 = 170, 170/2 = 85 (arithmetic mean of the quizzes' grades)

Repeating this process for each category, we would get 85, 90.5, and 78 for quizzes, tests, and term papers respectively. Now, we multiply each arithmetic mean by each percentage that makes up each subject. For instance, since quizzes make up 20% of the final grade, we would multiply 85 x 0.2 to get a final percentage of 100%. The percentage we multiply by is the weight value (w). Different weight values imply different importance and consideration. Note that we multiply by 0.2 since 0.2 is equivalent to 20%.

After applying this process three times for each subject, we would get 17%, 31.675%, and 35.1%. Summing these up, we arrive at a terminal weighted average grade of 83.775%. In contrast, the arithmetic mean would yield 84.5%. Note that the difference between the straight average (also known as the arithmetic mean) and the weighted average is minimal; weighted averages have a weight value to offset data points while straight averages lack this feature.

Geometric mean

The geometric mean is used to find a value "midway between two values" where our perception of the change between them isn't linear. Light, sound, and vibrations (as from an earthquake) work this way. For example, a 900 watt light bulb doesn't seem 9 times as bright as a 100 watt bulb. Thus, if you had a 500 watt bulb (the arithmetic average), it wouldn't seem halfway between the two, while a 300 watt light bulb (the geometric mean), would.

Advanced Measures
In the Advanced Measures unit, all measures will be referred to as: $$x_i\,\!$$,and the mean will be referred to as: $$\bar x\,\!$$.

In $$x_i\,\!$$, the $${}_i\,\!$$ just differentiates the different values of $$x\,\!$$, which there could be 10, 20, 30, as many as you want or are given by the problem. Also used in Advanced measures is the sum symbol, which looks like this: $$\sum\,\!$$. Always, there will be a number on top of it, below it, to the right of it, and sometimes to the left of it, like so: $$\sum_{i=1}^nx_i\,\!$$. All that that complicated symbol means is: "The sum ($$\sum\,\!$$) of $$x_i\,\!$$ ($$\sum x_i\,\!$$) with $$i\,\!$$ from 1 ($$\sum_{i=1}x_i\,\!$$) to n, which total, makes $$\sum_{i=1}^nx_i\,\!$$.

Sounds complicated, right? Wrong. All that long jumble means is that you add up all the values you have. Now, to get the mean, all you have to do is divide that total by however many numbers you have. So you get:

$$\bar x=\frac{\sum_{i=1}^nx_i}{n}\,\!$$

If there is a frequency, then:

$$\bar x=\frac{\sum_{i=1}^nf_ix_i}{n}\,\!$$

Now, you can start making a table of all known statistics:

Using $$\sum_{i=1}^nf_ix_i\,\!$$ and $$n\,\!$$, you can find $$\bar x\,\!$$ with the above formula, and add that to the chart:

So, lets put some measures in and see if you can figure it out: Exercise 1 Exercise 2 Exercise 3

Mean Deviation
The mean deviation is a measure that increases as the scores are farther away from the mean. It is mainly used to tell how dispersed the grades are: whether the mean value is every single value, or whether they jump around a lot and it somehow came to the mean. The formula for the mean variation is:

Mean Variation = $$\frac{\sum_{i=1}^n\left\vert x_i-\bar x\right\vert}{n}\,\!$$

(with frequency) $$\frac{\sum_{i=1}^nf_i\left\vert x_i-\bar x\right\vert}{n}\,\!$$

So, now we need to add new columns to our chart:

To Be Continued


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Primary mathematics/Percentages

Introduction
Percent, or per cent, or per 100 - cent meaning 100, as in century or a centipede is just telling you it's a number over 100. A percentage is a value divided by 100, shown with the percent symbol (%). For example, 4% is the same as the decimal value 0.04 or the fraction 4/100 (which could also be reduced to 1/25). If you write out the fraction or % as a long division problem, the way one finds the decimal value becomes clear.

4/100 --> 4% --> .04 (0.04)

Uses for percentages
Traditionally, percentages are used when dealing with changes in a value, particularly money. For example, a store may have a 20% off sale or a bank may charge 7.6% interest on a loan.

Defining the base
The base of a percentage change is the starting value. Many errors occur from using the wrong base in a percentage calculation.

Example
If a store has an $100 item, marks it 20% off, then charges 6% sales tax, what is the final price ?

First, calculate 20% of $100. That's 0.20 × $100 or $20. We then subtract that $20 from the original price of $100 to get a reduced sale price of $80.

Now we add in the sales tax. Here's where the tricky part comes in; what is the base ? That is, do we pay 6% tax on the original $100 price or on the reduced $80 price ? In most places, we would pay tax on the reduced sales price, so $80 is the base. Thus, we multiply $80 × 0.06 to get $4.80 and add that to $80 to get $84.80 for a final price. We can also multiply $80 * 1.06 to obtain the final price.

Notice that even though we took off 20% and then added back in 6%, this is not the same as taking off 14%, since the 20% and the 6% figures each had a different base. If the 6% sales tax did apply to the original full price, however, then both percentages would have the same base and the total reduction in price would, indeed, be 14%, bring the price down from $100 to $86.

Terms used with percentages

 * If you take 20% off an amount (or a 20% reduction), that means the new price is 20% less than the original (100%) price, so it's now 80% of the original price.

 * If you apply 20% interest to an amount, that means the new price is 20% higher than the original (100%) price, or 120% of the original price. (Note that this is simple interest, we will consider compound interest next.)

Compound interest
Simple interest is when you apply a percentage interest rate only once.

Compound interest is when you apply the same percentage interest rate repeatedly.

For example, let's say a $1000 deposit in a bank earns 10% interest each year, compounded annually, for three years. After the first year, $100 in interest will have been earned for a total of $1100. In the second year, however, there is not only interest on the $1000 deposit, but also on the $100 interest earned perviously. This "interest on your interest" is a feature of compounding. So, in year two we earn 10% interest on $1100, for $110 in interest. Add this to $1100 to get a new total of $1210. The 10% interest in the third year on $1210 is $121, which gives us a total of $1331.

For those familiar with powers and exponents, we can use the following formula to calculate the total:

T = P x (1 + I)N

Where:

T = final Total P = initial Principal I = Interest rate per compounding period N = Number of compounding periods

In our example, we get:

T = $1000 x (1 + 10%)3 = $1000 x (1 + 0.10)3 = $1000 x (1.10)3 = $1000 x (1.10 x 1.10 x 1.10) = $1000 x (1.331) = $1331

More complex calculations involving compound interest will be covered in later lessons.

title: Primary mathematics/Percentages


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Primary mathematics/Probability

Children and Probability
In mathematics probability tells us how likely something is to happen.

If something is absolutely certain to happen, we say its probability is one. If something cannot possibly happen, we say its probability is zero. A probability between zero and one means we don't know for sure what will happen, but the higher the number the more likely something is to happen. If something has a probability of 0.25, most people would say "it probably won't happen".

Unfortunately, children have a hard time understanding what a probability of 0.25 suggests. Children are much more adept at understanding probability in terms of percentages (25%), and fractions (i.e., "It will happen about ¼ of the time"), so it is very important that connections between probability and the development of those skills and understandings occur together.

Models
Probability is best taught with models. Two of the most effective and commonly used models used to teach probability are area and tree diagrams

Area diagrams give students an idea of how likely something is by virtue of the amount of space reserved for that probability. Consider the toss of a coin:

Students can imagine that every time they toss a coin, they need to put it into the appropriate side of the rectangle. As the number of throws increases, the amount of area needed to enclose the coins will be the same on both sides. The same probability can be shown using a tree diagram:

In this simple tree diagram, each line (or event path) will be followed ½ of the time. Note that both diagrams show all of the possibilities, not just the probability of one event.

Both of these models can become more complex. Consider all of the possibilities of throwing two coins in series:

In this area rectangle the instances where heads has been tossed twice in a row is represented in the upper left hand corner. Note that whereas the probability of tossing both heads and tails (in any order) is 50% or 1/2, and the probability of tossing two heads in a row is 25% or 1/4.

While the tree diagram gives us the same information, the student needs to understand that all of the events at the bottom have an equal chance of occurring (This is more visually evident in an area diagram). However, the tree diagram has the advantage of better showing the chronology of events (in this case the order of events is represented by downward motion).

While it may appear that both of these models have strengths and weaknesses, that is not the point. By becoming familiar with these and other models, students gain a more robust and diverse understanding of the nature of probability.

Consider the following example problem:

If at any time a Dog is just as likely to give birth to a male puppy as she is to give birth to a female puppy, and she has a litter of 5 puppies, what is the probability that all of the puppies will be of the same sex? While this problem can be easily answered with the a standard formula, $$P = 2 \left ( \frac{1}{2} \right ) ^n$$, where n is the number of puppies born, this nomenclature is not only unintelligible to students, but teaching it to students gives them no understanding of the underlying concepts. On the other hand, if they have explored problems similar to this with a tree diagram, they should fairly easily be able to see patterns emerging from which they can conjecture the "formula" that will find the correct answer. Consider that a student has drawn a tree diagram that shows the permutations with three puppies in the litter:

If a teacher was to have this student share their work, the class of students in looking at this model would be able to see that there are 8 possible outcomes. They should also notice that the only instances where litters of all males or all females can be found are on the far ends of the tree.

Students are always encouraged to look for patterns in many of their mathematical explorations. Here, students start to notice that the number of possibilities can be determined by multiplying the number 2 by itself exactly the number of times there is to be a new outcome (birth). In this case there are 3 outcomes (puppies born) in a row. So 2 x 2 x 2 = 23, or 8 possible outcomes. Once students start to make this connection from seeing this pattern, they may conjecture that if there are 5 puppies in a litter, the possible number of outcomes is 25, or 32, of which only 2 (represented by the combinations MMMMM, and FFFFF on the extreme ends) will result in a litter puppies of all the same sex.

Teachers should always be looking for students to make connections with other skills they are currently learning. In this case the teacher might ask the students to state the answer as a reduced fraction (1/16) and a percentage (6.25%).

title: Primary mathematics/Probability


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Primary mathematics/Factors and primes

Factors are factorizations, which is to say the dividing functions of a number. 100 = 10 x 10 88 = 8 x 11

Prime factors are factors which are prime numbers 55 = 5 x 11 100 = 2 x 5 x 2 x 5

Because 2, 5, and 11 cannot be divided by any number but 1 and itself, these numbers are called prime. Prime numbers are an important part of mathematics, and the problem of factoring large numbers into primes remains fundamental to the study of computer science. Factoring a number like 64 often requires more than one step:

64 = 2 x 32 (32 is not prime) or = 2 x 2 x 16 (16 is not prime) or = 2 x 2 x 2 x 8 (8 is not prime) or = 2 x 2 x 2 x 2 x 4 (4 is not prime) or = 2 x 2 x 2 x 2 x 2 x 2

title: Primary mathematics/Factors and primes


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Primary mathematics/Method for factoring

A method of prime factorization for children
In math, the way you get to the correct answer is just as important as getting the right answer. Sometimes the way one gets the answer can be thought of as a proof. People don't usually use the word "proof" when talking about arithmetic problems, but it's a good idea and good habit to get into. This is a method of prime factorization that constructs a proof that your answer is correct as you arrive at the answer. It keeps you from getting lost and also shows your work. More importantly, it helps you understand the process. It starts out very easy, so that beginners can use it, but later on lets you skip some steps to save time.

The basic point of prime factorization is to take a number and find the prime factors. Since each prime factor may occur more than once, for example in the number 4, which has prime factors (2,2), we know that the factors of a number are not a set but a multi-set. (The difference will be important as you get to more advanced math.) So the basic way we are going to do our work looks like this:

pf(X) = {a reason we believe this} (a list of factors)

For example, for the number 4, we could factorize it like this:

pf(4) = {4 = 2×2}   (2,2)

This is a very simple proof, and also represents what is sometimes called "showing our work" or "scratch paper". Learning math is often about learning the language and symbols of math, and we need to do that now.

Use the symbol pf(X) to mean "the prime factors of X". Use a comma separated list of numbers of "pf(X)" symbols between parentheses to mean a multi-set of prime factors. So (pf(45),2,5) means "multi-set that includes the prime-factors of 45 and the prime number 2 and the prime number 5".

Now, for doing prime factorization, we are only going to allow certain reasons to go from one step to the next toward the answer. For example, "my friend Harold said so" is not a good reason.

The first reason that we will allow is a simple multiplication equation of the form x = y×z. In particular, we are allowed to do this:

pf(x) = {x = y×z} (pf(y),pf(z))

so long as the equation x = y×z is true (and y and z are both greater than 1, otherwise we'd be going in circles!)

The second rule that we are allowed to use is to replace pf(x) with (x) if we know for sure that x is a prime. For example, you should probably memorize the first ten primes: (2, 3, 5, 7, 11, 13, 17, 19, 23, 29), so it's OK to substitute the number 23 for pf(23), since we should know by heart that 23 is prime. When we do this, we give the reason "23 is prime".

Finally, we need to note that ((X)) = (X), no matter what X is, so we never need to write parentheses inside parentheses (this is not true of multiplication, it is only true because we are using parentheses to mean something special.

So let's do an example:

pf(4) = {4 = 2×2}   (pf(2), pf(2) = {2 is prime}    (2,2)

So, in this example, we've chained together two steps, with a reason for each one that is very simple. This is a good idea when you are starting out. However, mathematicians also want to be efficient, so if you are sure of what you are doing, you could do two steps at once: pf(4) = {4 = 2×2, 2 is prime} (2,2)

Notice that whether we perform one, two, or even more steps, we are really creating an equation. If we drop out the intermediate steps, the result is pf(4) = (2,2), which is probably about what someone would want on a test.

Now let's do a harder one:

pf(60) = {60 = 6×10}   (pf(6), pf(10)) = {6 = 2×3, 10 = 2×5}   (2,3,2,5) = {commutativity of multiplication} (2,2,3,5)

Commutativity of multiplication is a rule that just lets us reorder the factors. It's nice to have them in order in our list, so we will often do that at the end. But that's a lot of writing, so we'll abbreviate it {com. of mult.} Also, notice that in the second step we dropped the unnecessary parentheses, instead of writing ((2,3), (2,5)).

Now let's do an even bigger one, that's still easy, since the prime factors are all small, and it's easy to see the factors.

pf(450) = {450 = 45×10}   (pf(45), pf(10)} = {45 = 5×9, 10 = 2×5}    (5,pf(9),2,5) = {9 = 3×3, com. of mult.}    (2,3,3,5,5)

Now we see a way in which we could have made a mistake --- if we forgot that 9 is not prime and had written (5,9,2,5). Hopefully we would have noticed, but we need to be careful!

When the factors aren't obvious
This is a nice method when the factors are easy to see. But, unfortunately, sometimes they aren't easy to see at all, and we need a new symbol for expressing our work in that case.

Consider the number 143. Is it prime, or does it have factors? Well, it's not obvious. The basic approach to finding any factor, or proving that a number is prime, is to start at the first prime (the number 2) and see if it's a factor. By going systematically up through each prime, we either find a factor or reach a prime that is bigger than the square root of the number, in which case we know the number is prime.

However, this is sometimes a big job, and we would really like to show our work in a way that helps us keep track of where we are and also can be quickly checked to make sure we haven't made any mistakes.

The way we can do this is with the symbol nu(X,Y), which means "the multi-set of prime factors of Y, understanding that none of the numbers up to and including X evenly divide Y". We're thus using the "not divisible up to" function nu both to help remember what has already been checked, but we've defined it so that the equation remains technically true at all times. The reason we want this symbol is that we know that is how you prove something is prime---you prove nu(sqrt(Y),Y), and then you know Y is prime. For example, if you know nu(13, 123), you know 123 is prime since 13×13 > 123. We can introduce nu(2,X) with the reason {X is not even}. If we know nu(2,X), then we can introduce nu(3,X) with the reason {sum of X's digits not divisible by 3}.

So let's try this:

pf(143) = { 143 is not even } (nu(2,143)) = { 1+4+3 not divisible by 3 } (nu(3,143)) = { 143 doesn't end in 5 or 0 } (nu(5,143)) = { 143 / 7 = 203/7 }   (nu(7,143)) = { 143 / 11 = 13!!}   (11,pf(13)) = { 13 is prime } (11,13)

So that is our answer. Now, in this case, we had to use a reason that we derived by division: 143/7 is not a whole number, but 203/7. And, we were surprised to learn that 11 evenly divides 143. Let's try an even harder one. (Note, we are using the divisibility test from the Dr. Math website, rather than doing division for many of these divisibility tests at noted, but if we had to do it on a test, we would use division, if we don't have those rules memorized.)

pf(1709) = { not even, digits don't sum to a multiple of 3, doesn't end in 5 or 0 } (nu(5,1709)) = { not divisible by 7 } (nu(7,1709)) = { not divisible by 11 } (nu(11,1709)) = { not divisible by 13 } (nu(13,1709)) = { not divisible by 17 } (nu(17,1709)) = { not divisible by 19 } (nu(19,1709)) = { not divisible by 23 } (nu(23,1709)) = { not divisible by 29 } (nu(29,1709)) = { not divisible by 31 } (nu(31,1709)) = { not divisible by 37 } (nu(37,1709)) = { not divisible by 41 } (nu(41,1709)) = { not divisible by 43, and 432>1709, so by nu rule we are done!} (1709)

By systematically working our way up from 2 to the square root of the number, we can always keep track of where we are. For example, if we had started with 6×1709 = 10254, we would have ended up with:

pf(10254) = { even } (2,pf(5127)) = {5+1+2+7=15 = 3×5 }   (2,3,pf(1709)) .....   (2,3,1709)

and then the proof would proceed as it did above, carrying with it the prime factors 2 and 3 the whole time.

This way of writing out your work is recommended. It works well for the child who will have trouble factorizing 54, and well for the child factorizing 10254. It builds good habits that will pay off in algebra and other more advanced mathematical calculation, and introduces, gently, the all-important concept of the proof.

One sometimes see factor trees suggested as a notation. Factor trees may be useful for visualizing some things but are not useful at all if one can't see the factors!


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User:Europol

de:Benutzer:Europol

title: User:Europol


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User:Checkoutmyhugeguns

Hi my name isCheckoutmyhugeguns my birthdate is 17 November 1982, 3:12 am.

title: User:Checkoutmyhugeguns


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Nonlinear finite elements/Kinematics - time derivatives and rates

Material time derivatives
Material time derivatives are needed for many updated Lagrangian formulations of finite element analysis.

Recall that the motion can be expressed as

\mathbf{x} = \boldsymbol{\varphi}(\mathbf{X}, t) \quad \text{or} \qquad \mathbf{X} = \boldsymbol{\varphi}^{-1}(\mathbf{x}, t) $$

If we keep $$\mathbf{X}$$ fixed, then the velocity is given by

\mathbf{V}(\mathbf{X}, t) = \frac{\partial \boldsymbol{\varphi}}{\partial t}(\mathbf{X}, t) $$ This is the  material time derivative expressed in terms of $$\mathbf{X}$$.

The spatial version of the velocity is

\mathbf{v}(\mathbf{x}, t) = \mathbf{V}(\boldsymbol{\varphi}^{-1}(\mathbf{x}, t), t) $$

We will use the symbol $$\mathbf{v}$$ for velocity from now on by slightly abusing the notation.

We usually think of quantities such as velocity and acceleration as spatial quantities which are functions of $$\mathbf{x}$$ (rather than material quantities which are functions of $$\mathbf{X}$$).

Given the spatial velocity $$\mathbf{v}(\mathbf{x}, t)$$, if we want to find the acceleration we will have to consider the fact that $$\mathbf{x} \equiv \mathbf{x}(\mathbf{X}, t)$$, i.e., the position also changes with time. We do this by using the chain rule. Thus

\cfrac{D\mathbf{v}(\mathbf{x},t)}{Dt} = \mathbf{a}(\mathbf{x}, t) = \frac{\partial \mathbf{v}(\mathbf{x},t)}{\partial t} + \frac{\partial \mathbf{v}(\mathbf{x},t)}{\partial \mathbf{x}}\cdot \frac{\partial \boldsymbol{\varphi}(\mathbf{X},t)}{\partial t}   = \frac{\partial \mathbf{v}}{\partial t} + \boldsymbol{\nabla} \mathbf{v}\cdot\mathbf{V} = \frac{\partial \mathbf{v}}{\partial t} + \boldsymbol{\nabla} \mathbf{v}\cdot\mathbf{v} $$ Such a derivative is called the  material time derivative expressed in terms of $$\mathbf{x}$$. The second term in the expression is called the  convective derivative..

Velocity gradient
Let the velocity be expressed in  spatial form, i.e., $$\mathbf{v}(\mathbf{x}, t)$$. The spatial velocity gradient tensor is given by

\boldsymbol{l} := \frac{\partial \mathbf{v}(\mathbf{x},t)}{\partial \mathbf{x}} = \boldsymbol{\nabla} \mathbf{v} $$ The velocity gradient $$\boldsymbol{l}$$ is a second order tensor which can expressed as

\boldsymbol{l} = l_{ij}~\mathbf{e}_i\otimes\mathbf{e}_j = \frac{\partial v_i}{\partial x_j}~\mathbf{e}_i\otimes\mathbf{e}_j $$ The velocity gradient is a measure of the relative velocity of two points in the current configuration.

Time derivative of the deformation gradient
Recall that the deformation gradient is given by

\boldsymbol{F} = \frac{\partial \boldsymbol{\varphi}}{\partial \mathbf{X}} $$ The time derivative of $$\boldsymbol{F}$$ (keeping $$\mathbf{X}$$ fixed) is

\dot{\boldsymbol{F}} = \frac{\partial }{\partial t}~\left(\frac{\partial \boldsymbol{\varphi}}{\partial \mathbf{X}}\right) = \frac{\partial }{\partial \mathbf{X}}~\left(\frac{\partial \boldsymbol{\varphi}}{\partial t}\right) = \frac{\partial \mathbf{v}}{\partial \mathbf{X}} = \boldsymbol{\nabla}_{\circ} \mathbf{v} $$ Using the chain rule

\dot{\boldsymbol{F}} = \frac{\partial \mathbf{v}}{\partial \mathbf{x}}\cdot\frac{\partial \mathbf{x}}{\partial \mathbf{X}} = \frac{\partial \mathbf{v}}{\partial \mathbf{x}}\cdot\frac{\partial \boldsymbol{\varphi}}{\partial \mathbf{X}} = \boldsymbol{l}\cdot\boldsymbol{F} $$ Form this we get the important relation

\boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} ~. $$

Time derivative of strain
Let $$d\mathbf{X}_1$$ and $$d\mathbf{X}_2$$ be two infinitesimal material line segments in a body. Then

d\mathbf{x}_1 = \boldsymbol{F}\cdot d\mathbf{X}_1 ~; d\mathbf{x}_2 = \boldsymbol{F}\cdot d\mathbf{X}_2 $$ Hence,

d\mathbf{x}_1\cdot d\mathbf{x}_2 = (\boldsymbol{F} \cdot d\mathbf{X}_1) \cdot (\boldsymbol{F} \cdot d\mathbf{X}_2) = d\mathbf{X}_1 \cdot (\boldsymbol{F}^T \cdot \boldsymbol{F}) \cdot d\mathbf{X}_2 = d\mathbf{X}_1 \cdot \boldsymbol{C} \cdot d\mathbf{X}_2 = d\mathbf{X}_1 \cdot (2~\boldsymbol{E} + \boldsymbol{\mathit{1}}) \cdot d\mathbf{X}_2 $$ Taking the derivative with respect to $$t$$ gives us

\frac{\partial }{\partial t} (d\mathbf{x}_1 \cdot d\mathbf{x}_2) = d\mathbf{X}_1 \cdot \frac{\partial \boldsymbol{C}}{\partial t} \cdot d\mathbf{X}_2 = 2~d\mathbf{X}_1 \cdot \frac{\partial \boldsymbol{E}}{\partial t} \cdot d\mathbf{X}_2 $$

The  material strain rate tensor is defined as

\dot{\boldsymbol{E}} = \frac{\partial \boldsymbol{E}}{\partial t} = \frac{1}{2}~\frac{\partial \boldsymbol{C}}{\partial t} = \frac{1}{2}~\dot{\boldsymbol{C}} $$ Clearly,

\dot{\boldsymbol{E}} = \frac{1}{2}~\frac{\partial }{\partial t}(\boldsymbol{F}^T\cdot\boldsymbol{F}) = \frac{1}{2}~(\dot{\boldsymbol{F}}^T \cdot \boldsymbol{F} + \boldsymbol{F}^T\cdot\dot{\boldsymbol{F}}) ~. $$

Also,

\frac{1}{2}~\frac{\partial }{\partial t}(d\mathbf{x}_1\cdot d\mathbf{x}_2) = d\mathbf{X}_1 \cdot \dot{\boldsymbol{E}} \cdot d\mathbf{X}_2 = (\boldsymbol{F}^{-1}\cdot d\mathbf{x}_1) \cdot \dot{\boldsymbol{E}} \cdot(\boldsymbol{F}^{-1}\cdot d\mathbf{x}_2) = d\mathbf{x}_1 \cdot (\boldsymbol{F}^{-T}\cdot \dot{\boldsymbol{E}} \cdot \boldsymbol{F}^{-1}) \cdot d\mathbf{x}_2 $$

The  spatial rate of deformation tensor or  stretching tensor is defined as

\boldsymbol{d} = \boldsymbol{F}^{-T}\cdot \dot{\boldsymbol{E}} \cdot\boldsymbol{F}^{-1} = \frac{1}{2}~\boldsymbol{F}^{-T}\cdot \dot{\boldsymbol{C}} \cdot\boldsymbol{F}^{-1} $$ In fact, we can show that $$\boldsymbol{d}$$ is the symmetric part of the velocity gradient, i.e.,

\boldsymbol{d} = \frac{1}{2}~(\boldsymbol{l} + \boldsymbol{l}^T) $$ For rigid body motions we get $$\boldsymbol{d} = \boldsymbol{\mathit{0}}$$.

Lie derivatives
Most of the operations above can be interpreted as push-forward and pull-back operations. Also, time derivatives of these tensors can be interpreted as Lie derivatives.

Recall that the push-forward of the strain tensor from the material configuration to the spatial configuration is given by

\boldsymbol{e} = \phi_{*} [\boldsymbol{E}] = \boldsymbol{F}^{-T}\cdot\boldsymbol{E}\cdot\boldsymbol{F}^{-1} $$ The pull-back of the spatial strain tensor to the material configuration is given by

\boldsymbol{E} = \phi^{*} [\boldsymbol{e}] = \boldsymbol{F}^T \cdot \boldsymbol{e} \cdot \boldsymbol{F} $$

Therefore, the rate of deformation tensor is a push-forward of the material strain rate tensor, i.e.,

\boldsymbol{d} = \boldsymbol{F}^{-T}\cdot\dot{\boldsymbol{E}}\cdot\boldsymbol{F}^{-1} = \phi_{*}[\dot{\boldsymbol{E}}] $$ Similarly, the material strain rate tensor is a pull-back of the rate of deformation tensor to the material configuration, i.e.,

\dot{\boldsymbol{E}} = \boldsymbol{F}^T \cdot \boldsymbol{d} \cdot \boldsymbol{F} = \phi^{*} [\boldsymbol{d}] $$

Now,

\boldsymbol{E} = \phi^{*}[\boldsymbol{e}] \quad \implies \quad \dot{\boldsymbol{E}} = \frac{\partial }{\partial t} \left(\phi^{*}[\boldsymbol{e}]\right) $$ Also,

\boldsymbol{d} = \phi_{*}[\dot{\boldsymbol{E}}] = \phi_{*} \left[\frac{\partial }{\partial t} \left(\phi^{*}[\boldsymbol{e}]\right)\right] $$ Therefore the rate of deformation tensor can be obtained by first pulling back $$\boldsymbol{e}$$ to the reference configuration, taking a material time derivative in that configuration, and then pushing forward the result to the current configuration.

Such an operation is called a  Lie derivative. In general, the Lie derivative of a spatial tensor $$\mathbf{g}$$ is defined as

\mathcal{L}_{\phi}[\boldsymbol{g}] := \phi_{*} \left[\frac{\partial }{\partial t} \left(\phi^{*}[\boldsymbol{g}]\right)\right] ~. $$

Spin tensor
The velocity gradient tensor can be additively decomposed into a symmetric part and a skew part:

\boldsymbol{l} = \frac{1}{2}~(\boldsymbol{l} + \boldsymbol{l}^T) + \frac{1}{2}(\boldsymbol{l} - \boldsymbol{l}^T) = \boldsymbol{d} + \boldsymbol{w} $$ We have seen that $$\boldsymbol{d}$$ is the rate of deformation tensor. The quantity $$\boldsymbol{w}$$ is called the  spin tensor.

Note that $$\boldsymbol{d}$$ is symmetric while $$\boldsymbol{w}$$ is skew symmetric, i.e.,

\boldsymbol{d} = \boldsymbol{d}^T ~; \boldsymbol{w} = -\boldsymbol{w}^T ~. $$

So see why $$\boldsymbol{w}$$ is called a "spin", recall that

\boldsymbol{l} = \dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} $$ Therefore,

\boldsymbol{w} = \frac{1}{2}(\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} - \boldsymbol{F}^{-T}\cdot\dot{\boldsymbol{F}}^T) $$ Also,

\boldsymbol{F} = \boldsymbol{R}\cdot\boldsymbol{U} \quad \implies \quad \dot{\boldsymbol{F}} = \dot{\boldsymbol{R}}\cdot\boldsymbol{U} + \boldsymbol{R}\cdot\dot{\boldsymbol{U}} $$ Therefore,

\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1} = (\dot{\boldsymbol{R}}\cdot\boldsymbol{U} + \boldsymbol{R}\cdot\dot{\boldsymbol{U}})\cdot (\boldsymbol{U}^{-1}\cdot\boldsymbol{R}^T) = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \boldsymbol{R}\cdot\dot{\boldsymbol{U}}\cdot\boldsymbol{U}^{-1}\cdot\boldsymbol{R}^T $$ and

\boldsymbol{F}^{-T}\cdot\dot{\boldsymbol{F}}^T = (\boldsymbol{R}\cdot\boldsymbol{U}^{-1})\cdot (\boldsymbol{U}\cdot\dot{\boldsymbol{R}}^T + \dot{\boldsymbol{U}}\cdot\boldsymbol{R}^T) = \boldsymbol{R}\cdot\dot{\boldsymbol{R}}^T + \boldsymbol{R}\cdot\boldsymbol{U}^{-1}\cdot\dot{\boldsymbol{U}}\cdot\boldsymbol{R}^T $$ So we have

\boldsymbol{w} = \frac{1}{2}~(\dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \boldsymbol{R}\cdot\dot{\boldsymbol{U}}\cdot\boldsymbol{U}^{-1}\cdot\boldsymbol{R}^T        - \boldsymbol{R}\cdot\dot{\boldsymbol{R}}^T - \boldsymbol{R}\cdot\boldsymbol{U}^{-1}\cdot\dot{\boldsymbol{U}}\cdot\boldsymbol{R}^T) $$ Now

\boldsymbol{R}\cdot\boldsymbol{R}^T = \boldsymbol{\mathit{1}} \quad \implies \quad \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \boldsymbol{R}\cdot\dot{\boldsymbol{R}}^T = \boldsymbol{\mathit{0}} $$ Therefore

\boldsymbol{w} = \dot{\boldsymbol{R}}\cdot\boldsymbol{R}^T + \frac{1}{2}~\boldsymbol{R}\cdot(\dot{\boldsymbol{U}}\cdot\boldsymbol{U}^{-1} -             \boldsymbol{U}^{-1}\cdot\dot{\boldsymbol{U}})\cdot\boldsymbol{R}^T $$ The second term above is invariant for rigid body motions and zero for an uniaxial stretch. Hence, we are left with just a rotation term. This is why the quantity $$\boldsymbol{w}$$ is called a spin.

The spin tensor is a skew-symmetric tensor and has an associated axial vector $$\boldsymbol{\omega}$$ (also called the angular velocity vector) whose components are given by

\boldsymbol{\omega} = \begin{bmatrix} w_1 \\ w_2 \\ w_3 \end{bmatrix} $$ where

\mathbf{w} = \begin{bmatrix} 0 & -w_3 & w_2 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0 \end{bmatrix} $$ The spin tensor and its associated axial vector appear in a number of modern numerical algorithms.

Rate of change of volume
Recall that

dv = J~dV \qquad \text{where}~ J = \det\boldsymbol{F} $$ Therefore, taking the material time derivative of $$dv$$ (keeping $$\mathbf{X}$$ fixed), we have

\cfrac{d}{dt}(dv) = \dot{J}~dV = \cfrac{\dot{J}}{J}~dv $$

At this stage we invoke the following result from tensor calculus:

If $$\boldsymbol{A}$$ is an invertible tensor which depends on $$t$$ then

\cfrac{d}{dt}(\det\boldsymbol{A}) = (\det\boldsymbol{A})~\text{tr}\left(\cfrac{d\boldsymbol{A}}{dt}\cdot\boldsymbol{A}^{-1}\right) $$

In the case where $$\boldsymbol{A} = \boldsymbol{F},~ J = \det\boldsymbol{F}$$ we have

\cfrac{d}{dt}(J) = J~\text{tr}\left(\dot{\boldsymbol{F}}\cdot\boldsymbol{F}^{-1}\right) $$ or,

\dot{J} = J~\text{tr}(\boldsymbol{l}) = J~\text{tr}(\mathbf{d}) $$ Therefore,

\cfrac{d}{dt}(dv) = \text{tr}(\mathbf{d})~dv $$ Alternatively, we can also write

\dot{J} = \frac{1}{2}~J~\boldsymbol{C}^{-1}:\dot{\boldsymbol{C}} $$ These relations are of immense use in numerical algorithms - particularly those which involved incompressible behavior, i.e., when $$\dot{J} = 0$$.


text: Continuum mechanics/Motion and displacement


text: Continuum mechanics/Strains and deformations


text: Continuum mechanics/Polar decomposition

title: Continuum mechanics/Polar decomposition


text: Continuum mechanics/Spectral decomposition


text: Continuum mechanics/Volume change and area change


text: Continuum mechanics/Time derivatives and rates


text: Continuum mechanics/Balance of mass

title: Continuum mechanics/Balance of mass


text: Continuum mechanics/Balance of linear momentum


text: Continuum mechanics/Balance of angular momentum


text: Continuum mechanics/Balance of energy

title: Continuum mechanics/Balance of energy


text:
Template:Clickurl

[    ]

This template attempts to superimpose an invisible link on an image. It may not work as expected for all browsers, and creates accessibility problems.

title: Template:Clickurl


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User talk:Europol

Hello Europol, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 22:26, 28 August 2007 (UTC)

Checkuser discussions
Hi Europol. If you can spare the time, your opinion would be greatly appreciated in the discussion currently going on about getting local Checkuser permissions here on Wikiversity. As an active user here, you're probably familiar with the "quiet, behind the scenes" way that we introduce new Custodians, but requesting Checkuser actually requires a "show of support" for the foundation stewards to give us this tool.

If you're not familiar with Checkuser, it's a tool we can use to find the source IP address (and/or alternate accounts) of vandals and others who mean harm to the project. This allows us to both "stop the problem at the source", or in some cases lets us know we can't stop something at the source if an IP is shared by one or more constructive contributors. --SB_Johnny | talk 11:02, 31 August 2007 (UTC)

title: User talk:Europol


text:
Template talk:Clickurl

Is this the same as Template:Click? --JWSchmidt 22:28, 28 August 2007 (UTC)

title: Template talk:Clickurl


text:
MediaWiki:Watchlist-details

NaN $1 pagess watched not counting talk pages.

title: MediaWiki:Watchlist-details


text:
Primary mathematics/Overview

What This Course is Not

This course assumes that the reader has reached a certain level of competency with the topics it covers. As such, this course is not designed to teach math to the reader. Rather, it is meant to give the reader insight into how and why mathematical core understandings and skills are taught the way they are. A basic mathematics background should be all that is needed to find value in this course. Often times parents (and even teachers) only understand a topic in the way they were taught it, but modern math curriculums are typically constructed such that topics are covered in many different ways in order to accommodate the different learning styles of students. There are many ways to teach any given mathematical understanding, and the depth of students' mathematical understandings are enhanced when they have explored it from multiple perspectives. For example, the Theory of Multiple Intelligences is but one perspective that teachers employ when leveraging student's individual strengths. This course will not teach the reader in step by step fashion how to multiply two large numbers, but the various building blocks and algorithms that students are introduced to in the process of learning this skill will be explained. Activities will be suggested that will present teachers and prospective home-schoolers with ideas for their own course content.

Because there are many learning styles, this course recognizes that there are many teaching styles. As such, this course is not necessarily intended to be a math curriculum, although it could serve as part of the foundation of one. There are many very good texts/curriculums/approaches out there; there are even "schools" of thought that suggest that math is best taught without textbooks if the teacher is skilled enough.

Connectedness

One of the overarching ideas that is highly valued in the teaching of mathematics in the primary grades involves the connections students make in their learning. All of our mathematical understandings are intertwined. For this reason different skills such as multiplication and concepts like those learned from the study of geometry are not taught separately. Rather, they are taught together, in such a way as to reinforce each other. Students should be encouraged to make connections by looking for patterns, exploring extremes, and forming and testing conjectures. Teachers and parents should refrain from simply "telling them how to do it." Letting student "guess at" and then check their work, learning from their mistakes as they go, leads to much stronger mathematical understandings. It also leads them to form mathematical habits that make future mathematically based explorations more efficient.

Primary and even upper school teachers recognize that understanding mathematics in the abstract is not the goal of most students. They need to see connections to the real world that inspire their learning. For this reason, teachers prefer to use real world problems that require the need for mathematical models (see below). Teachers prefer not to "teach". They prefer to "guide". Modern Educators realize that students gain true "ownership" of their understandings through, inasmuch as it is possible, making connections on their own - by way of their own work and explorations. Internal links in this course, when found in this course will purposefully be placed there to emphasize what should be an everpresent concept of the connectedness of mathematical understandings.

Manipulatives and Models

One very important component of the contemporary methods used to teach math is the use of manipulatives (such as toys) and models (visual representations) to give added dimension to students' understandings. In each chapter of this course, the reader will find various examples of models used to teach different mathematical understandings. Often, they will find that these models serve to make connections to material covered in other sections. Keep in mind that teachers in the classroom tend to be very creative and resourceful. Often, the models found in this course have many possible permutations, and can come in various shapes, sizes, and guizes. They are purposely presented here in simple forms to facilitate for the reader their identification.

title: Primary mathematics/Overview


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User talk:Leightwing

Hello Leightwing, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
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If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 01:37, 29 August 2007 (UTC)

making Wikiversity more welcoming to pre-college content
You might be interested in Pre-tertiary portal. What do you think about a portal or two that would serve pre-college students? I'd like to change the Main Page to include links to such portals (draft).

Checkuser discussions
Hi Leightwing. If you can spare the time, your opinion would be greatly appreciated in the discussion currently going on about getting local Checkuser permissions here on Wikiversity. As an active user here, you're probably familiar with the "quiet, behind the scenes" way that we introduce new Custodians, but requesting Checkuser actually requires a "show of support" for the foundation stewards to give us this tool.

If you're not familiar with Checkuser, it's a tool we can use to find the source IP address (and/or alternate accounts) of vandals and others who mean harm to the project. This allows us to both "stop the problem at the source", or in some cases lets us know we can't stop something at the source if an IP is shared by one or more constructive contributors. --SB_Johnny | talk 11:00, 31 August 2007 (UTC)

title: User talk:Leightwing


text:
User talk:Baltazart

Hello Baltazart, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
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If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 06:02, 29 August 2007 (UTC)

P.S. Have fun with our PHP resources.

title: User talk:Baltazart


text:
Lunar Boom Town/Building and construction facility

Building & Construction facility

Any town is constantly growing. So it needs qualified builders & domestic engineers, which will be capable of bringing any business up. The engineers will inevitably have a need to have a place, where orders are considered and planned, architects prepare projects, where builders and workers are hired, where the building machinery, cranes, tools, spacesuits etc. are stored. In fact this business should be established first to support rapid construction of other facilities.

Building on moon has certain advantages, as 6 times lighter gravity provides much less power requirements for lifting machinery, less requirements for construction stiffness etc. Any supporting construction can be finely built from aluminum, or some other cheap and unsuitable for earth construction materials.

From the other side - the lack of atmosphere gives much stricter requirements for building places, where human presence is necessary. Very hardy constructions and reliable systems must be implemented in any new building, which should contain regenerated and heated air. Buildings, suited for human beings also require all sorts of domestic communications, like water, heating, maybe common canalization, food delivery, transportation of some kind(possibly underground) and more. From this point of view - the other businesses on moon are best suited(financially) for robotic beings.

Successful facility work depends on
-materials
 * Lunar_Boom_Town/Steel_Production
 * Lunar_Boom_Town/Begillion_resource_harvesting

-machinery maintenance -electricity production -Oxygen Lunar_Boom_Town/LOX_Domestic_and_Exports -Fabrication methods Advanced_Automation_for_Space_Missions

Support Systems

 * Robotic cranes

Popular Construction Approaches

 * /Land prefab and bury
 * /Dig in and remodel
 * /Assemble and relocate
 * /Onsite fabrication


text:
BCP/Artemisia vulgaris

Artemisia vulgaris

title: BCP/Artemisia vulgaris


text:
BCP/Artemisia vulgaris/Logs

This page contains recent bloom logs for the plant Artemisia vulgaris (Mugwort).

Recent Logs

 * --SB_Johnny | PA! 10:46, 29 August 2007 (UTC)
 * --SB_Johnny | talk 09:21, 1 September 2007 (UTC)
 * SBJ 18:24, 4 September 2007 (UTC)
 * --SB_Johnny | talk 09:53, 15 September 2007 (UTC)

Archived Logs
(If there are more than 5 signatures under the recent logs heading, move the oldest logs to under this heading (generally leave the 3 most recent).

 * (no archived logs yet)

title: BCP/Artemisia vulgaris/Logs


text:
BCP/Chenopodium album

Chenopodium album

title: BCP/Chenopodium album


text:
BCP/Chenopodium album/Logs

2007 Logs

 * --SB_Johnny | PA! 11:28, 23 August 2007 (UTC)
 * --SB_Johnny | talk 09:58, 1 September 2007 (UTC)
 * --Jomegat 16:04, 12 September 2007 (UTC)
 * --SB_Johnny | talk 09:25, 27 September 2007 (UTC)
 * --SB_Johnny | talk 08:18, 1 October 2007 (UTC)
 * SB Johnny-NHAMP 09:21, 9 October 2007 (UTC)
 * --SB_Johnny | talk 12:21, 24 October 2007 (UTC)
 * --SB_Johnny | talk 17:27, 1 November 2007 (UTC)
 * --SB_Johnny | talk 10:34, 17 November 2007 (UTC)

2008 Logs

 * --Jomegat 00:09, 11 June 2008 (UTC)
 * --Jomegat 17:42, 17 June 2008 (UTC)
 * --Jomegat 02:30, 2 July 2008 (UTC)
 * --Jomegat 03:00, 8 July 2008 (UTC)
 * Jomegat 02:43, 15 July 2008 (UTC)
 * --SB_Johnny | talk 10:13, 20 July 2008 (UTC)
 * Jomegat 00:51, 2 August 2008 (UTC)
 * Jomegat 02:32, 13 August 2008 (UTC)
 * --SB_Johnny talk 13:40, 17 August 2008 (UTC)
 * --SB_Johnny talk 09:50, 23 September 2008 (UTC)
 * Jomegat 17:55, 7 October 2008 (UTC)

Recent Logs

 * Jomegat 04:08, 10 June 2009 (UTC)
 * Jomegat 00:23, 15 July 2009 (UTC)
 * Jomegat 00:47, 22 July 2009 (UTC)

title: BCP/Chenopodium album/Logs


text:
User talk:Elmedio~enwikiversity

Hello Elmedio, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
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If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! --JWSchmidt 15:54, 29 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called Elmedio. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name Elmedio~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:35, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:07, 19 April 2015 (UTC)

title: User talk:Elmedio~enwikiversity


text:
User talk:God~enwikiversity

Hello new user,

thank you for the feedback that Wikiversity is amazing. I have put a template in your edit: Learn. Do you know already: Username ?

"Wikiversity does not allow potentially inflammatory or offensive user names....Names of religious figures such as "God", "Jehovah", "Buddha", or "Allah", which may offend other people's beliefs"

""Please pick a username that is helpful in a learning environment. That means picking a name that you're comfortable writing under, but it also means picking a name that others are comfortable seeing and collaborating with."

You can read more below. We would like to see you around Wikiversity with another name, what do you think ? Erkan Yilmaz (evaluate me!, discussion ) 16:54, 29 August 2007 (UTC)

 * Hi. I'm afraid I agree with Erkan. We would very much like it if you chose another user name. Please tell us when you have done so, so that we can close this one down. Thank you, and hope to see you active at Wikiversity! McCormack 17:19, 29 August 2007 (UTC)

 * Hi - I can change your username for you, once you've decided one that you like. Just leave me a message on my talk page. Cheers, Cormaggio talk 11:25, 31 August 2007 (UTC)

Welcome
Hello God, and welcome to Wikiversity! If you need help, feel free to visit my talk page, or contact us and ask questions. When leaving comments for others to read, remember to sign and date; it helps everyone keep track of who is writing messages. The signature icon in the edit window makes it simple. To help you get started:
 * Take a guided tour and learn how to edit;
 * Explore our learning projects;
 * Browse our portals, schools, and research pages; and
 * Read and help develop our community policies.

If you're already excited about editing, our sandbox is the perfect place for all test edits. Also, don't forget to experiment with the links to your left. Be bold, and see you around Wikiversity! Erkan Yilmaz (evaluate me!, discussion ) 16:54, 29 August 2007 (UTC)

Your account will be renamed
Hello,

The developer team at Wikimedia is making some changes to how accounts work, as part of our on-going efforts to provide new and better tools for our users like cross-wiki notifications. These changes will mean you have the same account name everywhere. This will let us give you new features that will help you edit and discuss better, and allow more flexible user permissions for tools. One of the side-effects of this is that user accounts will now have to be unique across all 900 Wikimedia wikis. See the announcement for more information.

Unfortunately, your account clashes with another account also called God. To make sure that both of you can use all Wikimedia projects in future, we have reserved the name God~enwikiversity that only you will have. If you like it, you don't have to do anything. If you do not like it, you can pick out a different name.

Your account will still work as before, and you will be credited for all your edits made so far, but you will have to use the new account name when you log in.

Sorry for the inconvenience.

Yours, Keegan Peterzell Community Liaison, Wikimedia Foundation 23:37, 17 March 2015 (UTC)

Renamed
 This account has been renamed as part of single-user login finalisation. If you own this account you can |log in using your previous username and password for more information. If you do not like this account's new name, you can choose your own using this form after logging in: . -- Keegan (WMF) (talk) 06:09, 19 April 2015 (UTC)

title: User talk:God~enwikiversity


text:
Portal talk:Thelema

=What should be added?= What should this Wikiversity of Thelemic Studies add to the internet that the Thelema series on Wikipedia does not? Should it include...

 * summary of the philosophy of Thelema?
 * ramifications of Law of Thelema?
 * a compendium of links on different material?
 * quotations from Crowley and other contemporary Thelemites?
 * specific classes created (like certain other Wikiversities like Buddhist)?

=Welcome= Hey IAO131 23:13, 12 February 2008 (UTC)

Hey y'all I started up this page here to bring another layer to the versity. Discussion and expansion is encouraged. - User:Magosgruss 2:09 CST 8-29-07

=Topics= I definitely think that there should be some type of classes or working material to enrich the minds of visitors. Also, I would like to see a reference to the vision of the star sponge on the front page. A mention of rabelais and other saints wouldn't hurt. I am going to create a tasks page as well. Magosgruss 14:38 CST, 02.28.2008

 * What topics would the classes be on? The Star-sponge vision could be cited as an experiential example of the Thelemic cosmology of Nuit (infinite black space) and Hadit (but what twinkles!). Rabelais and others could be mentioned as possible antecedents of Thelema. 169.233.121.232 03:27, 29 February 2008 (UTC)

 * I think the classes could be in reference to cosmology and Thelemic Theory. Thanks for the contributions anonymous member. Magosgruss 04:00 02.29.2008 (UTC)
 * Oops, that was me forgetting to log in. What format do you think the classes should be in? Send a link of an example of a Wikiversity department you like/would like to emulate (at least at first) IAO131 03:17, 1 March 2008 (UTC)

 * http://en.wikiversity.org/wiki/What_Is_Strategy%3F_Why_Study_Strategy%3F is a good example of a well put together class. Magosgruss, 15:16, 3 March 2008

Community Resource
93 All, I am currently embarking on a project to get various OTO bodys as well as other thelemic organizations to upload to here online versions of the classes they teach so that we may all cross pollinate our knowledge base. magosgruss 17:51 02.29.2008 (UTC)

 * Here is a resource for some essays that might provide good class material. We could set it up in terms of difficult - perhaps have Active thelema at the 'beginning', 'psychological commentary' etc. IAO131 03:17, 1 March 2008 (UTC)

 * I added a link to active thelema, I think perhaps there should be a discourse as to the nature of the will along with it. I will construct some courses based on the rest of your essays as well.  I think some of the things in JoTS might be worth developing further like erwin hessle's articles on ethics. Magosgruss 05:40 03/01/2008 (UTC)

 * *Agreed re: the necessity of talking about Will, and Hessle's essay on Ethics. He did a great job at both summarizing Thelemic ethics and attacking some major misconceptions. I already linked his essay under Ethics > Resources, and my own response which sort of expands on Thelemic ethics called 'Thelemic Values: a new view of morality.' Check it out on the Wiki page IAO131 06:47, 1 March 2008 (UTC)

Discussion
93 all, I just had this idea to create some way on here to foster discussions of topics related to Thelema or maybe there is a better place for such groups? Not sure, but universities are often the center of intellectual progress. Anyway, perhaps we could establish some way of posting open ended questions and then discussing them to try to reach new understandings in subjects related to Thelema. magosgruss 03/15/2008 21:33 (UTC)
 * Wiki is not particularly suited for discussional material beyond this discussion page... Feel free to create discussion topics on here IAO131 04:12, 24 March 2008 (UTC)

title: Portal talk:Thelema


text:
Adobe Photoshop/Converting a Color Image to Black & White

A Black and White photograph is not a grayscale photograph, or a desaturated photograph.

In order to convert a color digital image to Black and White each color channel (red, blue and green) Must be converted individually. In Adobe Photoshop CS4 and above there is an automated process for this, using other editing software a Black and White image must first be made for each color channel (RGB) and layered together to form a true B&W photograph. Citation Citation

Below are Methods for creating a grayscale image.

Method #1

 * 1) Open the photo you want to convert.
 * 2) Choose Image > Mode > Grayscale
 * 3) When asked if you want to discard color info, click OK. Photoshop converts the colors in the image to black, white, and shades of gray. (this is called a grayscale image)

Method #2
Use Photoshop's Desaturate command.

A color image is considered saturated with color. When you desaturate an entire image you remove the color and create a grayscale image.

 * 1) Open the photo you want to convert.
 * 2) Choose Image > Adjustments > Desaturate  or
 * 3) Use the keyboard shortcut: "Shift+Ctrl+U" to Desaturate the image.

One advantage of the Desaturate command is the ability to perform this command on a specific area of the image. For instance, you can use the Marquee Tool to select the part of the image to be desaturated (grayscale) while leaving the rest of the image in color (or whatever the original was).


text:
Talk:Adobe Photoshop/Converting a Color Image to Black & White

How about using/mentioning also other editing tools ?
Hi, this seems to be specific to Photoshop. How about mentioning this also at begin if so ? Perhaps you could do the same steps with another open source / freeware tool, then everybody can do as described. Erkan Yilmaz (evaluate me!, discussion ) 19:39, 29 August 2007 (UTC)