text:
Geometry for Elementary School/Pythagorean theorem

In this chapter, we will discuss the Pythagorean theorem. It is used to find the side lengths of a right triangles. It says:
 * In any right triangle, the area of the square whose side is the hypotenuse (the side of a right triangle opposite the right angle) is equal to the sum of areas of the squares whose sides are the two legs (i.e. the two sides other than the hypotenuse).

This means that if $$\triangle ABC $$ is a right triangle, the length of the hypotenuse, c, squared eqauls the sum of a squared plus b squared. Or:
 * $$a^2 + b^2 = c^2 \,$$

Here's an example:

In a right-angled triangle, a=5cm and b=12cm, so what is c?

 * $$a^2 + b^2 = c^2 \,$$
 * $$c = \sqrt{ 5^2 + 12^2 }\,$$
 * $$c = \sqrt{ 25 + 144 }\,$$
 * $$c = \sqrt{ 169 }\,$$
 * $$c = 13 \,$$

If c is not larger than a or b, your answer is incorrect. There may be a number of reasons that your answer is incorrect. The first is that you have calculated the sums wrong, the second is that the triangle you are trying to find the hypotenuse of is not a right angled triangle or the third is you have mixed up the measurements. There may be more finer points to having a wrong answer but the three stated are the most common

Exercise

 * Write about Pythagorean Theorum and its use to prove that $$\sqrt{0.5}$$ is an irrational number.

Geometria per scuola elementare/Il Teorema di Pitagora


text:
Geometry for Elementary School/Constructing a triangle

In this chapter, we will show how to construct a triangle from three segments. The construction is based on Book I, proposition 22

The construction
Given three line segments $$\overline{AB}$$, $$\overline{CD}$$ and $$\overline{EF}$$ we build a triangle whose sides equal the segments.

 * 1) Copy the line \overline{CD} to point A. GEOM constr triangle 01a.png If you have forgotten how to do this, follow the instructions in the previous section. Your construction should look like the grey lines in the picture below. Call the new line $$\overline{AG}$$  GEOM constr triangle 01b.png It's a good idea to erase your construction lines now, so all that's left are the four line segments shown below.  GEOM constr triangle 01c.png
 * 2) Copy the line $$\overline{EF}$$ to point B GEOM constr triangle 02a.png  Your construction should look like the grey lines in the picture below. Call the new line $$\overline{BH}$$ [[image:GEOM constr triangle 02b.png|300 px]]
 * 3) Draw the circle $$\circ A,\overline{AG}$$, whose center is A and radius is $$\overline{AG}$$.
 * 4) Draw the circle $$\circ B,\overline{BH} $$, whose center is B and radius is $$\overline{BH}$$.
 * 5) Let J be an intersection point of $$\circ A,\overline{AG} $$ and $$\circ B,\overline{BH} $$. GEOM constr triangle 03 first.png
 * 6)  Draw a line $$\overline{AJ}$$.
 * 7)  Draw a line $$\overline{BJ}$$. GEOM constr triangle 03.png

Claim
The sides of the triangle $$\triangle ABJ$$ equal to $$\overline{AB}$$, $$\overline{CD}$$ and $$\overline{EF}$$.

Proof

 * 1) The segment $$\overline{AB}$$ is a side of the triangle and equal to itself.
 * 2) The segment $$\overline{AJ}$$ is equal to $$\overline{AG}$$ because they are both radii of circle$$\circ A,\overline{AG}$$. And because it was copied, $$\overline{AG}$$=$$\overline{CD}$$. Therefore $$\overline{AJ}$$ is also equal to $$\overline{CD}$$
 * 3) The segment $$\overline{BJ}$$ is equal to $$\overline{BH}$$ because they are both radii of circle$$\circ B,\overline{BH}$$. And because it was copied, $$\overline{BH}$$=$$\overline{EF}$$. Therefore $$\overline{BJ}$$ is also equal to $$\overline{EF}$$
 * 4) Hence the sides of the triangle $$\triangle ABJ$$ are equal to $$\overline{AB}$$, $$\overline{CD}$$ and $$\overline{EF}$$.

Testing the procedure

 * 1) Draw a line $$\overline{AB}$$ of some length.
 * 2) Copy the line $$\overline{AB}$$ to an arbitrary point C and get $$\overline{CD}$$.
 * 3) Draw a line $$\overline{EF}$$ such that it length is three times the length of $$\overline{AB}$$. (We didn't specify how to construct such a segment and we give it as an exercise. Use chapter Copy the line as a guide for the solution.
 * 4) Construct a triangle from $$\overline{AB}$$, $$\overline{CD}$$ and $$\overline{EF}$$.

Why you couldn't construct the triangle in the test?
The reason we couldn’t build the triangle in the test is that the circles we constructed did not intersect. One cannot use any three segment to construct a triangle. The length of the segments must obey a condition called “The triangle inequality”. The triangle inequality states that any of the segments should be shorter that the sum of the length of the other two segments. If one of the segments is longer the circles do not intersect. If one segment equals to the sum of the other two, we get a line instead of a triangle.

Therefore, the construction is correct but one should condition the segments on which it can be applied. Note that the original construction was conditioned by Euclid, hence there is no error in the construction or in its proof.

Geometria per scuola elementare/Costruzione di un triangolo

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


text: User talk:80.44.90.178

title: User talk:80.44.90.178


text:
A-level Mathematics/OCR/D1

Please post all questions to A-level Mathematics/Discussion.

Contents

 * 1) /Algorithms/
 * 2) /Node Graphs/
 * 3) /Linear Programming/

title: A-level Mathematics/OCR/D1


text:
A-level Mathematics/OCR/D1/Node Graphs

Contents

 * 1) /Introduction/
 * 2) /Spanning Trees/
 * 3) /Shortest Path Algorithms/
 * 4) /Bipartite Graphs & Matching/
 * 5) /Route Inspection/
 * 6) /Travelling Salesperson Problem/

title: A-level Mathematics/OCR/D1/Node Graphs


text:
Talk:Calculus/Tables of Derivatives

Trying to make the layout of this page similar to the table of integrals.

title: Talk:Calculus/Tables of Derivatives


text:
Calculus/Table of Trigonometry

Definitions

 * $$\tan(x)=\frac{\sin(x)}{\cos(x)}$$
 * $$\sec(x)=\frac{1}{\cos(x)}$$
 * $$\cot(x)=\frac{\cos(x)}{\sin(x)}=\frac{1}{\tan(x)}$$
 * $$\csc(x)=\frac{1}{\sin(x)}$$

Inverse trigonometric functions

 * $$\arcsin(x) = \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t= -i\log(ix + \sqrt{1-x^2})$$
 * $$\arccos(x) = \frac{\pi}{2} - \arcsin(x) = \frac{\pi}{2} - \int_0^x \frac{1}{\sqrt{1-t^2}}\mathrm{d}t = \frac{\pi}{2} + i\log(ix + \sqrt{1-x^2})$$
 * $$\arctan(x) = \int_0^x \frac{1}{1+t^2} \mathrm{d}t = \frac{i}{2}\log\left(\frac{1-ix}{1+ix}\right)$$
 * $$\arccsc(x) = \arcsin\left(\frac{1}{x}\right) = -i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)$$
 * $$\arcsec(x) = \arccos\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arcsin\left(\frac{1}{x}\right) = \frac{\pi}{2} + i\log\left(\frac{i}{x} + \sqrt{1 - \frac{1}{z^2}}\right)$$
 * $$\arccot(x) = \arctan\left(\frac{1}{x}\right) = \frac{\pi}{2} - \arctan(x) = \frac{\pi}{2} + \frac{i}{2}\log\left(\frac{1+ix}{1-ix}\right)$$
 * $$\arcsin(x) + \arcsin(y) = \arcsin\left(x\sqrt{1-y^2} + y\sqrt{1-x^2}\right)$$
 * $$\arccos(x) + \arccos(y) = \arccos\left(xy - \sqrt{(1-x^2)(1-y^2)}\right)$$
 * $$\arctan(x) + \arctan(y) = \arctan\left(\frac{x+y}{1-xy}\right) \pmod \pi$$

Pythagorean Identities

 * $$\sin^2(x)+\cos^2(x)=1$$
 * $$1+\tan^2(x)=\sec^2(x)$$
 * $$1+\cot^2(x)=\csc^2(x)$$

Double Angle Identities

 * $$\sin(2x)=2\sin(x)\cos(x)$$
 * $$\cos(2x)=\cos^2(x)-\sin^2(x)$$
 * $$\tan(2x)=\frac{2\tan(x)}{1-\tan^2(x)}$$
 * $$\cos^2(x)=\frac{1+\cos(2x)}{2}$$
 * $$\sin^2(x)=\frac{1-\cos(2x)}{2}$$

Angle Sum Identities

 * $$\sin(x+y)=\sin(x)\cos(y)+\cos(x)\sin(y)$$

 * $$\sin(x-y)=\sin(x)\cos(y)-\cos(x)\sin(y)$$

 * $$\cos(x+y)=\cos(x)\cos(y)-\sin(x)\sin(y)$$

 * $$\cos(x-y)=\cos(x)\cos(y)+\sin(x)\sin(y)$$

 * $$\sin(x)+\sin(y)=2\sin\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$

 * $$\sin(x)-\sin(y)=2\cos\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$

 * $$\cos(x)+\cos(y)=2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right)$$

 * $$\cos(x)-\cos(y)=-2\sin\left(\frac{x+y}{2}\right)\sin\left(\frac{x-y}{2}\right)$$

 * $$\tan(x)+\tan(y)=\frac{\sin(x+y)}{\cos(x)\cos(y)}$$

 * $$\tan(x)-\tan(y)=\frac{\sin(x-y)}{\cos(x)\cos(y)}$$

 * $$\cot(x)+\cot(y)=\frac{\sin(x+y)}{\sin(x)\sin(y)}$$

 * $$\cot(x)-\cot(y)=\frac{-\sin(x-y)}{\sin(x)\sin(y)}$$

Product-to-sum identities

 * $$\cos(x)\cos(y)= \frac{\cos(x+y)+\cos(x-y)}{2}$$

 * $$\sin(x)\sin(y)=\frac{\cos(x-y)-\cos(x+y)}{2}$$

 * $$\sin(x)\cos(y)=\frac{\sin(x+y)+\sin(x-y)}{2}$$

 * $$\cos(x)\sin(y)=\frac{\sin(x+y)-\sin(x-y)}{2}$$

In terms of the complex exponential

 * $$e^{i\theta} = \mathrm{cis} \theta = i\sin \theta + \cos\theta$$
 * $$\sin\theta = \mathrm{Re}(e^{i\theta}) = \frac{e^{i\theta}-e^{-i\theta}}{2i}$$
 * $$\cos\theta = \mathrm{Im}(e^{i\theta}) = \frac{e^{i\theta}+e^{-i\theta}}{2}$$
 * $$\tan\theta = \frac{\sin\theta}{\cos\theta} = \frac{e^{2i\theta}-1}{i(e^{2i\theta}+1)}$$
 * $$\csc\theta = \frac{1}{\sin\theta} = \frac{2i}{e^{i\theta}-e^{-i\theta}}$$
 * $$\sec\theta = \frac{1}{\cos\theta} = \frac{2}{e^{i\theta}+e^{-i\theta}}$$
 * $$\cot\theta = \frac{1}{\tan\theta} = \frac{i(e^{2i\theta}+1)}{e^{2i\theta}-1}$$

Hyperbolic functions

 * $$e^x = \sinh x + \cosh x$$
 * $$\cosh^2 x - \sinh^2 x = 1$$
 * $$\mathrm{sech}^2 x = 1 - \tanh^2 x$$
 * $$\mathrm{csch}^2 x = \mathrm{coth}^2 x - 1$$
 * $$\sinh x = -i\sin ix = \frac{e^{x}-e^{-x}}{2}$$
 * $$\cosh x = \cos ix = \frac{e^{x}+e^{-x}}{2}$$
 * $$\tanh x = -i\tan ix = \frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}$$
 * $$\mathrm{csch} x = i\csc ix = \frac{2}{e^{x}-e^{-x}}$$
 * $$\mathrm{sech} x = \sec ix = \frac{2}{e^{x}+e^{-x}}$$
 * $$\mathrm{coth} x = i\cot ix = \frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}$$

Inverses

 * $$\mathrm{arsinh} x = \int_0^x \frac{1}{\sqrt{t^2 + 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 + 1}\right)$$
 * $$\mathrm{arcosh} x = \int_1^x \frac{1}{\sqrt{t^2 - 1}} \mathrm{d}t = \log\left(x + \sqrt{x^2 - 1}\right)$$
 * $$\mathrm{artanh} x = \int_0^x \frac{1}{1-t^2} \mathrm{d}t = \frac{1}{2}\log\left(\frac{1+x}{1-x}\right)$$
 * $$\mathrm{arccsh} x = \log\left(\frac{1 + \sqrt{1 + x^2}}{x}\right)$$
 * $$\mathrm{arsech} x = \log\left(\frac{1 + \sqrt{1 - x^2}}{x}\right)$$
 * $$\mathrm{arcoth} x = \frac{1}{2}\log\left(\frac{x+1}{x-1}\right)$$

title: Calculus/Table of Trigonometry


text:
Climate Change/Mitigation Strategies/Sustainable Development

More than a decade after the Rio Summit in 1992, the challenge of moving economic development forwards, without compromising the availability of resources for future generations and at the same time protecting the environment and cultural heritage has become more urgent than ever.

Appearing on the scene with the publication of Limits to Growth, it was not until the 1987 Brundtland Report, produced by the World Commission on Environment and Development, that sustainable development entered the political arena. The idea was to identify a “pathway” through which people could create sustainable policies and practices rather than to develop a blueprint for action.

Ever since, the concept of sustainable development (SD) has managed to “stay alive” and has actually created much dispute and discussion. Since its emergence, SD has been widely embraced by government bodies and other influential organizations around the world, yet the actual behaviour of institutions that have claimed devotion to sustainability has been much criticized.

Sustainable development and sustainability have been used differently, but there is no evident consistency of difference. There has been much debate about whether and how the usages have differed or should differ, but these debates are unresolved. There is not even much agreement on which term is broader, or which carries more undesirable baggage. Here, the two terms are synonymous.

One of the drawbacks of SD as a concept has been the lack of agreement on its definition and guiding principles, although this is probably also a reason why the term has lasted for a long time and has become so widely accepted and debated. However one may choose to define it, sustainability stands as a critique; it is a challenge to prevailing assumptions, institutions and practices.

It relates to the basic non-viability of current development trends and practices and, therefore, it implies a need for significant change. Sustainability is indeed a challenge to conventional thinking. It requires the capacity to adapt to constantly changing conditions, as well as the flexibility to work with uncertainty, and with differences in local conditions and in public expectations shaped by culture, values and experience. Above all, it is participatory, ensuring that local communities and individuals have substantive input into designing and implementing development programs and projects.

Sustainability is more than setting up a system of compatible relations among biophysical, social, and economic aspects. It is also a matter of culture and governance. How relations are built with each other, what habits of thought and behaviour we establish and how we go about making decisions are central to sustainability.

It implies considering both the present and future, the incorporation of economic, environmental and social considerations into planning and management, the recognition that our institutions must evolve so that they can deal with the linkages and complexity of the world, the combination of technical and local knowledge systems, and to seek to change underlying values, beliefs and attitudes.


text:
Talk:US Internet Law/Section 230

Is it really InternetLaw or just Internet Law? 14:55, 2 October 2005 (UTC) this is pointless

title: Talk:US Internet Law/Section 230


text:
Talk:Guide to Unix/BSD

Transclusion by BSD
This page is not a real bookshelf, as it contains no Wikibooks. Instead, it contains external links. I wanted some external links for BSD, which is a submodule of Guide to UNIX, so I transcluded this page. Kernigh 15:14, 2 October 2005 (UTC)

Merge is complete. BSD bookshelf redirects to Guide to UNIX/BSD. Its talk page is now at Talk:Guide to UNIX/BSD. --Kernigh 19:36, 30 October 2005 (UTC)

ISBN numbers
I noticed that the OpenBSD 3.8 set is ISBN 0-9731791-6-3 and that wiki makes ISBN hyperlinks. This particular ISBN link seems useless, though. --Kernigh 19:36, 30 October 2005 (UTC)

title: Talk:Guide to Unix/BSD


text:
Wikijunior talk:Big Cats/Caracal

Hyraxes
What are hyraxes?

==Hdavis 15:48, 2 October 2005 (UTC)

Small, odd looking, squat little animals that live on rock formations and which are distantly related to the horse and hippopotamus. They are classed in Class Mammalia, Order Hyracoidea.

John Burkitt 17:30, 18 November 2005 (UTC)

title: Wikijunior talk:Big Cats/Caracal


text:
Wikijunior talk:Big Cats/Serval

Wording
"The head looks disproportionately large compared to the body, and the ears look very large."

I think that "disproportionately" should be changed to looks bigger, or something like that.

John B. Here: The legs are unusually long and skinny, the neck is unusually long and skinny. In fact the head looks rather small for a cat that size. The photo being used does not show off this distinguishing characteristic...that's like a picture of a skunk that doesn't show the white stripe down the back.... I'll have to check and see if I can find a suitable picture of our serval Streaker to upload. John Burkitt 17:34, 18 November 2005 (UTC)

 * Right now our photo sources are either wikipedians' zoo snapshots, U.S. National Fish and Wildlife photos, or from the very very few people who put their safari photos under a free license. It kinda limits the quality of our illustrations.  — Laura Scudder | Talk 18:04, 18 November 2005 (UTC)

 * Yes, you have to remember John that if you upload a file, it can be used by anyone else for basically anything. Just be aware of that. -- LV (Dark Mark) 18:16, 18 November 2005 (UTC)

 * I am not only aware of that, but educating people is one of the primary reasons I take animal photos. I won't have any sleepless nights over it.  John Burkitt 20:15, 20 November 2005 (UTC)

title: Wikijunior talk:Big Cats/Serval


text:
Canadian Property Law/Personal Property

Introduction
In the common law systems personal property may also be called chattels. It is distinguished from real property. Personal property may be classified in a variety of ways, such as goods, money, negotiable instruments, security, and intangible assets including choses in action.

Personality
Back to Canadian Property Law

title: Canadian Property Law/Personal Property


text:
Talk:Introduction to Philosophy/Zen Buddhism

I do not like this page, it seems sort of repetitive and I am not sure why the focus is on Zen, a particular sect of Buddhism instead of Buddhism in general, and like a quick paragraph long (or more) explanation of the 4 noble truths, or something like that. It just seems like a misprepresentation or something. I will try to come back to it myself if noone else does in a couple days, I am sorta busy at the moment

-bob

I'm not sure how to join in the discussion, I'm new to this so pardon any mistakes on my part. I have to agree with Bob on this one. Not only is the text very thin, it also strikes me as rather unrepresentative of Zen Buddhism itself. "Focus on nature and children"? Apart from the fact that the writer has quite obviously missed the point of zen teachings, I find it hard to believe any zen practitioner would subscribe to the ideas presented in the text. Also, shouldn't the different schools of thought be explained? The koans, the importance of zazen, the historical roots of the tradition? The inherent incommunicability of zen? In my opinion this text does not measure up to Wikipedia standards and should be deleted outright.

-Tom

This page seems more like a reference to stoicism than to Zen Buddhism. I certainly don't see much about Zen philosophy here that would not overlap with stoicism.

-whitecphil

Reference
The reference link is now dead. Malafaya (talk) 16:39, 29 April 2009 (UTC)


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Canadian Constitutional Law/Sources of Constitutional law

Introduction
Canada is a kingdom; Queen Elizabeth II is the Queen of Canada; all executive, legislative, and judicial power is vested in the Queen. These facts are established in the Constitution Act, 1867. However, in 1947 the King transferred all of his legal power to the Governor General (GG) of Canada, except for the power of appointing a GG. It might be said, then, that the GG occupies the top position in the government legally but that the Queen does so symbolically. Similarly, it might be said that all executive, legislative and judicial power is vested in the GG. The constitution, however, provides that these powers can be exercised only on the initiative of the ministers, the Parliament, or the judiciary, respectively.

The government is organized in three branches: executive; legislative; and judicial.

The executive branch consists of the GG, the Prime Minister, and the other cabinet ministers. This branch carries out the administration of government and prepares the governmental plans, budgets and legislative proposals for consideration by the legislative branch. In the administration of government, the executive acts under the authorization of either existing legislation or of the inherent legal power (the "prerogative") of the GG. To take an executive action, the Prime Minister or a minister advises the GG to take the action and (for routine executive business) the GG must comply. It might be said that the GG occupies the top position in the executive legally but that the Prime Minister does functionally. There is some evidence that one role of the GG is to refuse to act on advice that would give an advantage to one political party over another. Similarly, it might be said that the GG should refuse a proposal from the Prime Minister that would give himself/herself increased power or that would otherwise amend the constitution. The GG and the rest of the Executive Branch are subject to Canadian constitutional law.

The legislative branch consists of the Queen, the House of Commons, and the Senate. One purpose of the Commons is to install or replace the Executive, sometimes through a vote of non-confidence. Another purpose is to consider and accept, modify, or reject the plans, budget, and proposed legislation originated by the Executive. Parliamentary debates before confederation indicated that the first purpose of the Senate was to ensure that proposed legislation did not impose an unfair majority view on a minority geographical region; it might be said that the veto power of the Senate existed for this purpose. A second purpose was said to be the modification of any bill expressing an unwise but popular view of the Commons. A third purpose was to review for errors or omissions any bill passed by the Commons. (The first and second purposes of the Senate seem to have been forgotten now.) Representation in the Commons is by population; representation in the Senate, by geographical region, to offset the large legislative power of large provinces. The GG enacts legislation on the advice and consent of the Commons and the Senate; the GG may refuse to do this (presumably to protect the constitution, as for refusal to carry out an executive act on the advice of the Prime Minister) but has not refused for many decades.

Court judgments are given in the name of the Queen. Superior court judges are appointed for life, to promote independence. Federally, an independent public prosecutor has been set up to remove political influence on decisions of whether or not to prosecute in a particular case.

Since Canada began life as a British colony, its form of democracy is based on the model of the United Kingdom, a model that is known in Canada as Responsible Government,. Both countries share a constitutional monarchy and responsible government, in which the powers of the Executive are given to the party (or coalition) that has the confidence of the elected house of the legislature becomes installed in the executive branch and the GG acts on the advice of the Prime Minister and ministers. Many other aspects of Canada's public law derive from the United Kingdom.

However, Canada departed from Britain the British model from the outset by being a federal union in which legislative power is divided between a central parliament and provincial legislatures. The division of powers was set out in the UK British North America Act, now issued as the (Canadian) Constitution Act, 1867. Since each level of government's power was limited by the other, the courts reviewed legislation to determine whether it was constitutional, a role they have never played in the United Kingdom.

Provincial governments are organized in the same way as the federal under their Lieutenant Governor. They are as competent and independent in their field of jurisdiction as the federal government is in its field.

In 1982, Canada's constitution was fundamentally reformed by adding a judicially-enforced Canadian Charter of Rights and Freedoms (the "Charter"), a guarantee of the rights of Canada's indigenous peoples, and a constitutional amending formulas. The Charter has led to important changes to Canada's law and society, as well as to an ongoing debate about the relative role of judges and politicians in determining how Canadians should be governed. The amending formulas must be followed if the "entrenched" part of constitutional law is to be amended; other parts of the constitution can be amended without reference to the amending formulas.

What is the Constitution of Canada?
Canada's constitution defines itself, or at least it tries to. Section 52 (2) of the Constitution Act, 1982 says:

The Constitution of Canada includes (a) the Canada Act, 1982, including this Act; (b) the Acts and orders referred to in the schedule; and (c) any amendment to any Act or order referred to in paragraph (a) or (b).

The Canada Act is a reference to the statute of the UK Parliament that ended the British role in amending Canada's constitution. The Constitution Act, 1982 was appended to the Canada Act, and for practical purposes they are the same. Part I of the Constitution Act, 1982 is the Canadian Charter of Rights and Freedoms. Part II sets out the Rights of the Aboriginal Peoples of Canada. Part III is entitled Equalization and Regional Disparities and provides for a non-justiciable commitment to regional equalization. Parts IV and IV.1 provide for constitutional conferences. More importantly, Part V provides a Procedure for Amending the Constitution of Canada. Part VI amended the Constitution Act, 1867, Canada's original constitution. Part VII is General.

The Schedule referred to in paragraph (b) lists 30 historical documents that fill out Canada's written constitution. By far the most important is the first, the Constitution Act, 1867, which continues to be the primary source for the division of sovereignty between the federal and provincial governments. There have been some minor amendments since 1982, but the two major attempts at constitutional change, the Meech Lake Accord and the Charlottetown Accord, resulted in failure.

For most purposes, therefore, Canada's written constitution consists of the two documents, the Constitution Act, 1867 and the Constitution Act, 1982. However, as section 52 (2) recognizes by the use of the word "includes," Canada's constitution is not all written. Both the Constitution Act, 1867 and the Constitution Act, 1982 are written against a background of principles inherited from the United Kingdom and not set out explicitly in the text. Some of these unwritten elements are enforceable in court, and others are constitutional conventions, derived from tradition and enforced by the political system.

An example of an unwritten constitutional concept is the office of the prime minister. While the prime minister of is the most powerful official in the Canadian government, the Canadian written constitution hardly mentions the existence of a prime minister. No law says how the prime minister is chosen or what powers he or she exercise. These matters are governed by constitutional convention.

The Supreme Court of Canada summed up the "unwritten constitution" this way:
 * The Constitution is more than a written text. It embraces the entire global system of rules and principles which govern the exercise of constitutional authority. A superficial reading of selected provisions of the written constitutional enactment, without more, may be misleading. It is necessary to make a more profound investigation of the underlying principles animating the whole of the Constitution, including the principles of federalism, democracy, constitutionalism and the rule of law, and respect for minorities. (Re: Secession of Quebec [1998] 2 SCR 217 headnote)

Federalism
Canadian federalism means that the sovereignty of Canada is divided between two levels, or "orders" of government: federal and provincial. Each province has its own government, and the constitution gives the federal government certain powers and each provincial government certain powers. The division of powers is listed in Sections 91 and 92 of the Constitution Act, 1867. Section 91 lists the powers of the federal government, while section 92 lists the powers of the provincial governments.

The most important "unwritten" constitutional principle relating to federalism is that Canada has full sovereignty, that is, Canada can exercise all of the powers of an independent country. All of Canada's sovereignty can theoretically be exercised by either the federal or provincial governments; there is nothing that one or the other government cannot do. Another unwritten constitutional principle is that it is up to Canada's courts to decide which government has the power to do what, when the rules are unclear.

Democracy
Reading the Constitution Act, 1867, the most important Constitutional document until 1982, the impression one would get is that Canada is not a democracy, but a colony ruled directly by a Governor appointed by the Queen of England. An elected legislature exists, but all of the laws of the legislature must be approved by the Governor and most executive decisions are made by the Governor and his or her Council.

This is because one of the most important constitutional rules of Canada is unwritten. It is called Responsible Government. The important aspects of responsible government are:
 * The Governor must appoint his or her Council (aka. "Council of Ministers" or "Cabinet") from the political party with the most support in the legislature
 * The Governor must choose the leader of the political party with the most support in the legislature as Prime Minister
 * The Governor must act "on the advice of the Prime Minister" (i.e., the Governor must do whatever the Prime Minister says) or "on the advice of Council" (i.e., the Governor must do as instructed by Cabinet).

Therefore, whichever party wins the election gets the most seats in the legislature. Their party leader becomes Prime Minister. The Prime Minister tells the Governor who to appoint to the Cabinet, and then the Cabinet tells the Governor how to run the country. In this way, the Governor's position becomes completely ceremonial; sovereignty, which is technically exercised by the Queen through her Governor, is in fact exercised through elected officials.

Responsible government applies equally to provincial governments. Each province has a ceremonial Lieutenant-Governor, appointed by the Queen, who acts "on the advice" of the Premier (the provincial equivalent of the Prime Minister) and the provincial Cabinet.

Constitutionalism and the rule of law
Constitutionalism means that the Constitution is supreme. Statutes passed by legislatures may not violate the constitution; any part of a statute that violates the constitution is null. The government is not free to disobey the constitutional rules when it suits them, be they written or unwritten.

The rule of law is a fundamental principle, not only of Canadian constitutional law, but of Canadian life. The stereotype of Canadians as straight-laced people who want everything done "by the book" suggests how seriously the country takes the rule of law. The Rule of Law principle is that the government may not act arbitrarily, but must act only according to the authority given them by the law. The law must, in turn, be valid according to the following criteria:

 * 1) It must be validly approved (i.e., passed according to the rules of the legislature and given "Royal Assent" by the Governor)
 * 2) It must be consistent with the constitution
 * 3) It must be possible to obey  (this usually means that it cannot conflict with another law,  it must be made public, it cannot be retroactive, etc., although judges sometimes make exceptions to these rules)

Respect for Minorities
This constitutional principle has existed since the beginning of the Canadian legal system, although this has not always led to minorities being respected in practice. Nowadays, it is a principle that guides courts in interpreting things such as Aboriginal treaties and the place of French speakers in Canada.

The Royal Prerogative
The royal prerogative are the powers given to the crown by the common law. These powers are unique to the crown and have no equivalent power to any one else. In modern times, the royal prerogative has fallen into disuse. The courts have held that there is no longer the prerogative power to legislate or administer justice, and there remains little of this power left. The prerogative exists primarily as a convention and is generally accepted as no longer being enforced by the courts.

The Constitution Act, 1867 was not meant to be a complete constitution for an independent country. In 1867, the territory of what is now Canada was broken up into several provinces of the British Empire. The Constitution Act, 1867 was simply a statute passed by the British Parliament to federate several of its provinces. The relations between the Empire and its Canadian colonies would remain the same: the Empire would still run Canada's foreign affairs, sign and ratify treaties on its behalf, and declare war in its name.

When Canada became an independent country, the powers that the British Empire exercised on Canada's behalf fell to the federal government. However, they were not powers that the Canadian Parliament (legislature) inhereited. Rather, they were powers that the Governor-General inherited. As stated above, the powers of the Governor-General are in fact exercised by the Prime Minister. Therefore, decisions on how to run Canada's foreign affairs, to sign and ratify treaties, and declare war are not decisions that the Parliament of Canada make. Rather, they are made directly by the Prime Minister and Cabinet.

Constitution Act, 1867
The Constitution Act, 1867 was the product of negotiations between the top officials of four British colonies in North America: Canada, Nova Scotia, New Brunswick, and Prince Edward Island. They were discussing ways to deal with several common problems: the huge expenses of building rail and canal projects, worsening relations with the United States, and the unworkable political systems that made the colonies hard to govern. In the end, the provinces of Canada, Nova Scotia, and New Brunswick agreed on a plan to create a federation of four provinces (by dividing the former province of Canada into the provinces of Ontario and Quebec). They proposed legislation to the Imperial Parliament in London, which passed it. On July 1, 1867, the British North America Act came into effect, creating a new British colony, "the Dominion of Canada" (s.3).

The highlights of the British North America Act (a.k.a. "the BNA Act") were:

 * 1)  A federal system: powers would be shared between a new federal government and a government in each province (s.91 and 92);
 * 2)  The federal government would have vast powers in the Canada's most crucial financial and economic areas (currency, all forms of taxation, the debt, banks, interest, the building of interprovincial railways and canals) as well as other areas of "national" concern, such as aboriginals and their lands, and criminal law (s. 91)
 * 3) The provincial governments would generally have powers over local matters and areas that affected the common people: property rights, licenses, hospitals, cities, courts, government land; (s. 92)
 * 4) The capital of the new country would be Ottawa, on the Quebec-Ontario border; (s.16)
 * 5) The federal government would have a House of Commons: an elected legislature with representation from the provinces roughly equal to the size of their population; (s.37-40)
 * 6) The federal government would also have a Senate, with 24 representatives each from Ontario and Quebec and 12 representatives each from New Brunswick and Nova Scotia, the members of which would be chosen by the Governor (in fact, by the Prime Minister of Canada); (s. 21-36)
 * 7) The provincial governments would each have one elected legislative assembly; Quebec would also have an appointed Legislative Council; (s.69, 71, 88)
 * 8) The federal government could set up a general court of appeal for Canada and other courts for the "better administration of the laws of Canada", and appoint the judges; (s. 101)
 * 9) While the provinces were in charge of courts, the Governor (i.e., Prime Minister of Canada) had the power to appoint the judges to the most important provincial courts. (s.96)

Back to Constitutional law


text:
Talk:Mirad

Untitled
I disagree with the recent change to the representation of the pronunciation of Unilingua "a".

The previous page had "a in English 'rah!'" which corresponds to /A/ in SAMPA for General American, and /A:/ in British Received Pronunciation. The revision shows "a in English 'talk'" which is a much lower vowel, /O/ in General American and /O:/ in RP, and comes closer to the pure O sound in Spanish "no" than to A.

In my dialect of American English (Southern California mixed with Great Lakes), there is a clear distinction between "a" in "father" and "aw" in "awful". The recent change specifies the latter as the general Unilingua A, but this is not usually perceived as a "pure" or general-purpose A sound.

Remember that as an auxiliary language, Unilingua is subject to many different interpretations of its vowel sounds, and to a lesser extent its consonants. The page itself points out that "r" and "j" will have varying pronunciations depending on the origin of speakers.

I believe it is best to specify vowels that will hold up as distinct when a variety of regional accents is applied to them. That means replacing "talk" with "father" or some other comparatively "neutral A" word on the Unilingua page.

To a much lesser extent, I am also concerned about the change from "u in Spanish 'tu'" to "u in English 'flu'", since there are some English speakers who pre-yodify the vowel in "flu" so it sounds like "flyoo." However, I also understand the potential for confusion between Spanish "tu" and French "tu"; the latter is definitely not a generalized U sound! Perhaps English speakers who already pre-yodify U will have to adjust their pronunciation to accommodate Unilingua pre- and post-yodification (I have to do something similar to cope with a minimal-pair distinction between words like cin "picture" and ciyn "mark").

-- Doug Ewell, Fullerton, California http://users.adelphia.net/~dewell/

You guys should re-read carefully your entire book.
I've found some inconsistencies in your book. Things that just don't match, or disprove previously said statements. Some phrases and expressions you present here are merely word by word translation of their English counterparts. The greeting (cu baka) is probably influenced by the Russian "Zdravstvuite" (which virtually means the same as Unilinguian "cu baka" - be healthy). Same goes for some other inherently cultural expressions. Word "mir" in Russian means world, just like in Unilingua, and some words that derive from ber resemble Russian words with similar meanings.

The slogan on your main page (Glojo mir ebdayo zey yacana dayen) is also wrong.
Language according to your word families is dad. Why do you use dayen? I didn't find a word that anyway cognates with it, though it looks similar to daer. Is it meant to be noun derived from daer in instrumental case? Why not dadey (dad+ey)? To complicate things even further, in some languages adjectives should be inflected with same grammatical case as the noun they describe, is it so in Unilingua? You should specify that. I'm not sure if preposition zey (through, across) is appropriate in this specific expression, although it seems to be an instrumental preposition. Neither am I sure about its necessity (you've got cases, why use preposition?). I'd say your slogan should sound: Glojo mir ebdayo yacana dadey (or yacanaey dadey). In language with cases it is a good idea to present a table of prepositions sorted by cases. This way you may know what is the correct perposition to use in every grammatical case.

You need to make it all more consistent.
Revise it and rethink it. Rethink the idea of dropping original graphemes that English lacks (I know what pain can it be to write it, but one may produce keyboard layout for it, or a script that would reformat the yodification). All the yodification you use may be misleading, and may complicate the fresh students of this language. It has created complications for me. And I speak fluently three languages: Russian, Hebrew and English. They are of different linguistic groups, scripts, morphosintactic typology and have only a very small common vocabulary (mostly of somewhat religious words like halleluia and amen, and technical terms that derive from Latin). Even for me, with my knowledge of such diverse languages it was difficult to get used to the fact that things written like yoy, uy etc, are a single phoneme. As more phonemes have corresponding graphemes so better.

Now, guys you may curse me and spit on me for the critics I write, but...
Believe me I write it to help and not to annoy. I see the original author's idea as the idea of the language of thought. It won't be a language of thought if its spoken version is merely translation of words of some other language. It should use its own grammar and logics of expressions. Greeting word could be anything depending on your culture. In many cultures the word or expression used like that looses with time its original meaning. You should think about it carefully. The greeting could be some word that would mean glad to initiate conversation. In Unilingua it could be a combined in to a single word that has f, j, and d consonants (I guess). I want Unilingua to stay logical language.

P.S.
Are there any other sources on-line except this one for this language? Did you know that there is NO wikipedia article about Unilingua and Noubar Agapoff?!

Book rename
Should this book be renamed Mirad since it's not really on Unilingua, but a revision thereof? --Swift (talk) 02:04, 26 October 2008 (UTC)

title: Talk:Mirad


text:
Topology/Connectedness

Motivation
To best describe what is a connected space, we shall describe first what is a disconnected space. A disconnected space is a space that can be separated into two disjoint groups, or more formally:

A space $$(X,\mathcal{T})$$ is said to be disconnected iff a pair of disjoint, non-empty open subsets $$X_1, X_2$$ exists, such that $$ X = X_1 \cup X_2$$.

A space $$X$$ that is not disconnected is said to be a connected space.

Examples
A picture to illustrate:
 * 1) A closed interval $$[a,b]$$ is connected. To show this, suppose that it was disconnected. Then there are two nonempty disjoint open sets $$A$$ and $$B$$ whose union is $$[a,b]$$. Let $$X$$ be the set equal to $$A$$ or $$B$$ and which does not contain $$b$$. Let $$s=\sup X$$. Since X does not contain b, s must be within the interval [a,b] and thus must be within either X or $$[a,b]\setminus X$$. If $$s$$ is within $$X$$, then there is an open set $$(s-\varepsilon,s+\varepsilon)$$ within $$X$$. If $$s$$ is not within $$X$$, then $$s$$ is within $$[a,b]\setminus X$$, which is also open, and there is an open set $$(s-\varepsilon,s+\varepsilon)$$ within $$[a,b]\setminus X$$. Either case implies that $$s$$ is not the supremum.
 * 2) The topological space $$X = (0,1)\setminus\{{\frac{1}{2} } \}$$ is disconnected: $$A = (0,\frac{1}{2} ), B = (\frac{1}{2}, 1)$$

As you can see, the definition of a connected space is quite intuitive; when the space cannot be separated into (at least) two distinct subspaces.

Definitions
Definition 1.1

A subset $$U$$ of a topological space $$X$$ is said to be clopen if it is both closed and open.

Definition 1.2

A topological space X is said to be totally disconnected if every subset of X having more than one point is disconnected under the subspace topology

Theorems about connectedness
If $$X$$ and $$Y$$ are homeomorphic spaces and if $$X$$ is connected, then $$Y$$ is also connected.

Proof: Let $$X$$ be connected, and let $$f$$ be a homeomorphism. Assume that $$Y$$ is disconnected. Then there exists two nonempty disjoint open sets $$Y_1$$ and $$Y_2$$ whose union is $$Y$$. As $$f$$ is continuous, $$f^{-1}(Y_1)$$ and $$f^{-1}(Y_2)$$ are open. As $$f$$ is surjective, they are nonempty and they are disjoint since $$Y_1$$ and $$Y_2$$ are disjoint. Moreover, $$f^{-1}(Y_1) \cup f^{-1}(Y_2)=f^{-1}(Y)=X$$, contradicting the fact that $$X$$ is connected. Thus, $$Y=f(X)$$ is connected.

Note: this shows that connectedness is a topological property.

If two connected sets have a nonempty intersection, then their union is connected.

Proof: Let $$A$$ and $$B$$ be two non-disjoint, connected sets. Let $$X$$ and $$Y$$ be non-empty open sets such that $$X\cup Y=A\cup B$$. Let $$a_0\in A$$. Without loss of generality, assume $$a_0\in X$$.

As $$A$$ is connected, $$a\in X\forall a\in A$$ ...(1).

As $$Y$$ is non-empty, $$\exists b\in B$$ such that $$b\in Y$$.

Hence, similarly, $$b\in Y\forall b\in B$$ ...(2) Now, consider $$c\in A\cap B$$. From (1) and (2), $$c\in X\cap Y$$, and hence $$X\cap Y\neq\emptyset$$. As $$X,Y\in\mathcal{T}$$ are arbitrary, $$A\cup B$$ is connected.

If two topological spaces are connected, then their product space is also connected.''

Proof: Let X1 and X2 be two connected spaces. Suppose that there are two nonempty open disjoint sets A and B whose union is X1×X2. If for every x∈X, {x}×X2 is either completely within A or within B, then π1(A) and π1(B) are also open, and are thus disjoint and nonempty, whose union is X1, contradicting the fact that X1 is connected. Thus, there is an x∈X such that {x}×X2 contains elements of both A and B. Then π2(A∩{(x,y)}) and π2(B∩{(x,y)}), where y is any element of X2, are nonempty disjoint sets whose union is X2, and which are a union of open sets in {(x,y)} (by the definition of product topology), and are thus open. This implies that X2 is disconnected, a contradiction. Thus, X1×X2 is connected.

Exercises

 * 1) Show that a topological space $$X$$ is disconnected if and only if it has clopen sets other than $$\emptyset$$ and $$X$$ (Hint: Why is $$X_1$$ clopen?)
 * 2) Prove that if $$f:X\to Y$$ is continuous and surjective (not necessarily homeomorphic), and if $$X$$ is connected, then $$Y$$ is connected.
 * 3) Prove the Intermediate Value Theorem: if $$f:[a,b]\to \mathbb{R}$$ is continuous, then for any $$y$$ between $$f(a)$$ and $$f(b)$$, there exists a $$c\in [a,b]$$ such that $$f(c)=y$$.
 * 4) Prove that $$\mathbb{R}$$ is not homeomorphic to $$\mathbb{R}^2$$ (hint: removing a single point from $$\mathbb{R}$$ makes it disconnected).
 * 5) Prove that an uncountable set given the countable complement topology is connected (this space is what mathematicians call 'hyperconnected')
 * 6) a)Prove that the discrete topology on a set X is totally disconnected.

b) Does the converse of a) hold (Hint: Even if the subspace topology on a subset of X is the discrete topology, this need not imply that the set has the discrete topology)

title: Topology/Connectedness


text:
Neuroscience/Electrophysiology

Neurophysiology at Wikipedia

Neurophysiology historically began with Luigi Galvani in 1791 when he published the monograph "De Viribus Electricitatis in Motu Musculari Commentarius" (Commentary on the Effect of Electricity on Muscular Motion). This monograph suggested an inherent electricity exists within the living tissue.

The study of nervous system electrophysiology can occur at many different levels including: 1) Electroencephalography (EEG) 2) Local Field Potentials 3) neuron populations and 4) the electrical activity outside or inside a neuron measured by patch clamp methods. As such, the major focus of this field is on cellular communication. Frequently studied as the generation of electrical signals within neurons from the regulated flow of various ions, often in the form of an action potential. A good beginning reference is the "Physiology of the Nervous System" by David Ottoson.

This field has been intertwined with Computational Neuroscience the work of Andrew Huxley, Alan Hodgkin, and David Marr.

title: Neuroscience/Electrophysiology


text:
Topology/Path Connectedness

Definition
A topological space $$X$$ is said to be path connected if for any two points $$x_0, x_1\in X$$ there exists a continuous function $$f:[0,1]\to X$$ such that $$f(0)=x_0$$ and $$f(1)=x_1$$

Example
The preceding example works in any convex space (it is in fact almost the definition of a convex space).
 * 1) All convex sets in a vector space are connected because one could just use the segment connecting them, which is $$f(t)=t\vec{a}+(1-t)\vec{b}$$.
 * 2) The unit square defined by the vertices $$[0,0], [1,0], [0,1], [1,1]$$ is path connected. Given two points $$(a_0, b_0), (a_1,b_1)\in [0,1]\times[0,1]$$ the points are connected by the function $$f(t)=[(1-t)a_0+ta_1,(1-t)b_0+tb_1]$$ for $$t\in[0,1]$$.

Adjoining Paths
Let $$X$$ be a topological space and let $$a,b,c\in X$$. Consider two continuous functions $$f_1,f_2:[0,1]\to X$$ such that $$f_1(0)=a$$, $$f_1(1)=b=f_2(0)$$ and $$f_2(1)=c$$. Then the function defined by

$$f(x) = \left\{ \begin{array}{ll} f_1(2x) & \text{if } x \in [0,\frac{1}{2}]\\ f_2(2x-1) & \text{if } x \in [\frac{1}{2},1]\\ \end{array} \right.$$

Is a continuous path from $$a$$ to $$c$$. Thus, a path from $$a$$ to $$b$$ and a path from $$b$$ to $$c$$ can be adjoined together to form a path from $$a$$ to $$c$$.

Relation to Connectedness
Each path connected space $$X$$ is also connected. This can be seen as follows:

Assume that $$X$$ is not connected. Then $$X$$ is the disjoint union of two open sets $$A$$ and $$B$$. Let $$a\in A$$ and $$b\in B$$. Then there is a path $$f$$ from $$a$$ to $$b$$, i.e., $$f:[0,1]\rightarrow X$$ is a continuous function with $$f(0)=a$$ and $$f(1)=b$$. But then $$f^{-1}(A)$$ and $$f^{-1}(B)$$ are disjoint open sets in $$[0,1]$$, covering the unit interval. This contradicts the fact that the unit interval is connected.

Exercises
is connected but not path connected.
 * 1) Prove that the set $$A=\{(x,f(x))|x\in\mathbb{R}\}\subset\mathbb{R}^2$$, where $$f(x) = \left\{ \begin{array}{ll} 0 & \text{if } x \leq 0\\ \sin(\frac{1}{x}) & \text{if } x > 0\\ \end{array} \right.$$

title: Topology/Path Connectedness


text:
Introduction to Art/Painting Exercises

A white object on a white object
In Old Masters paintings, a white plate is quite distinct from the tablecloth on which it rests. Obviously, the eye perceives very clear and distinct differences between the two. But what are they? Painting a white object resting on a white object is an extreme exercise to train your eye to see the myriad colors that reflect off white objects.

title: Introduction to Art/Painting Exercises


text: User talk:24.222.241.178

title: User talk:24.222.241.178


text:
User:Redge

You can find my website at romkevandermeulen.nl.

title: User:Redge


text:
Calculus/Integration techniques/Trigonometric Substitution

The idea behind the trigonometric substitution is quite simple: to replace expressions involving square roots with expressions that involve standard trigonometric functions, but no square roots. Integrals involving trigonometric functions are often easier to solve than integrals involving square roots.

Let us demonstrate this idea in practice. Consider the expression $$\sqrt{1-x^2}$$. Probably the most basic trigonometric identity is $$\sin^2(\theta)+\cos^2(\theta)=1$$ for an arbitrary angle $$\theta$$. If we replace $$x$$ in this expression by $$\sin(\theta)$$, with the help of this trigonometric identity we see
 * $$\sqrt{1-x^2}=\sqrt{1-\sin^2(\theta)}=\sqrt{\cos^2(\theta)}=\cos(\theta)$$

Note that we could write $$\theta=\arcsin(x)$$ since we replaced $$x^2$$ with $$\sin^2(\theta)$$.

We would like to mention that technically one should write the absolute value of $$\cos(\theta)$$, in other words $$\bigl|\cos(\theta)\bigr|$$ as our final answer since $$\sqrt{a^2}=|a|$$ for all possible $$a$$. But as long as we are careful about the domain of all possible $$x$$ and how $$\cos(\theta)$$ is used in the final computation, omitting the absolute value signs does not constitute a problem. However, we cannot directly interchange the simple expression $$\cos(\theta)$$ with the complicated $$\sqrt{1-x^2}$$ wherever it may appear, we must remember when integrating by substitution we need to take the derivative into account. That is we need to remember that $$dx=\cos(\theta)d\theta$$, and to get a integral that only involves $$\theta$$ we need to also replace $$dx$$ by something in terms of $$d\theta$$. Thus, if we see an integral of the form
 * $$\int\sqrt{1-x^2}dx$$

we can rewrite it as
 * $$\int\cos(\theta)\cos(\theta)d\theta=\int\cos^2(\theta)d\theta$$

Notice in the expression on the left that the first $$\cos(\theta)$$ comes from replacing the $$\sqrt{1-x^2}$$ and the $$\cos(\theta)d\theta$$ comes from substituting for the $$dx$$.

Since $$\cos^2(\theta)=\frac{1+\cos(2\theta)}{2}$$ our original integral reduces to:
 * $$\tfrac12\int d\theta+\tfrac12\int\cos(2\theta)d\theta$$.

These last two integrals are easily handled. For the first integral we get
 * $$\tfrac12\int d\theta=\tfrac12\theta$$

For the second integral we do a substitution, namely $$u=2\theta,du=2d\theta$$ to get:
 * $$\tfrac12\int\cos(2\theta)d\theta=\tfrac12\int\cos(u)\tfrac12du=\tfrac14\sin(u)=\frac{\sin(2\theta)}{4}$$

Finally we see that:
 * $$\int\cos^2(\theta)d\theta=\frac{\theta}{2}+\frac{\sin(2\theta)}{4}=\frac{\theta+\sin(\theta)\cos(\theta)}{2}$$

However, this is in terms of $$\theta$$ and not in terms of $$x$$, so we must substitute back in order to rewrite the answer in terms of $$x$$.

That is we worked out that:
 * $$\sin(\theta)=x\qquad\cos(\theta)=\sqrt{1-x^2}\qquad\theta=\arcsin(x)$$

So we arrive at our final answer
 * $$\int\sqrt{1-x^2}dx=\frac{\arcsin(x)+x\sqrt{1-x^2}}{2}$$

As you can see, even for a fairly harmless looking integral this technique can involve quite a lot of calculation. Often it is helpful to see if a simpler method will suffice before turning to trigonometric substitution. On the other hand, frequently in the case of integrands involving square roots, this is the most tractable way to solve the problem. We begin with giving some rules of thumb to help you decide which trigonometric substitutions might be helpful.

If the integrand contains a single factor of one of the forms $$\sqrt{a^2-x^2}\mbox{ or }\sqrt{a^2+x^2}\mbox{ or }\sqrt{x^2-a^2}$$ we can try a trigonometric substitution.

 * If the integrand contains $$\sqrt{a^2-x^2}$$ let $$x=a\sin(\theta)$$ and use the identity $$1-\sin^2(\theta)=\cos^2(\theta)$$.
 * If the integrand contains $$\sqrt{a^2+x^2}$$ let $$x=a\tan(\theta)$$ and use the identity $$1+\tan^2(\theta)=\sec^2(\theta)$$.
 * If the integrand contains $$\sqrt{x^2-a^2}$$ let $$x=a\sec(\theta)$$ and use the identity $$\sec^2(\theta)-1=\tan^2(\theta)$$.

Sine substitution

If the integrand contains a piece of the form $$\sqrt{a^2-x^2}$$ we use the substitution
 * $$x=a\sin(\theta)\quad dx=a\cos(\theta)d\theta$$

This will transform the integrand to a trigonometric function. If the new integrand can't be integrated on sight then the tan-half-angle substitution described below will generally transform it into a more tractable algebraic integrand.

E.g., if the integrand is $$\sqrt{1-x^2}$$ ,

 * $$\begin{align}

\int\limits_0^1\sqrt{1-x^2}dx&=\int\limits_0^\frac{\pi}{2}\sqrt{1-\sin^2(\theta)}\cos(\theta)d\theta\\ &=\int\limits_0^\frac{\pi}{2}\cos^2(\theta)d\theta\\ &=\tfrac12\int\limits_0^\frac{\pi}{2}\bigl(1+\cos(2\theta)\bigr)d\theta\\ &=\frac{\pi}{4} \end{align}$$

If the integrand is $$\sqrt{\frac{1+x}{1-x}}$$, we can rewrite it as
 * $$\sqrt{\frac{1+x}{1-x}}=\sqrt{\frac{1+x}{1+x}\cdot\frac{1+x}{1-x}}=\frac{1+x}{\sqrt{1-x^2}}$$

Then we can make the substitution

 * $$\begin{align}

\int\limits_0^a\frac{1+x}{\sqrt{1-x^2}}dx&=\int\limits_0^\alpha\frac{1+\sin(\theta)}{\cos(\theta)}\cos(\theta)d\theta &0<a<1\\ &=\int\limits_0^\alpha\bigl(1+\sin(\theta)\bigr)d\theta&\alpha=\arcsin(a)\\ &=\alpha+\Big[-\cos(\theta)\Big]_0^\alpha\\ &=\alpha+1-\cos(\alpha)\\ &=1+\arcsin(a)-\sqrt{1-a^2} \end{align}$$

Tangent substitution

When the integrand contains a piece of the form $$\sqrt{a^2+x^2}$$ we use the substitution
 * $$x=a\tan(\theta)\qquad\sqrt{x^2+a^2}=a\sec(\theta)\qquad dx=a\sec^2(\theta)d\theta$$

E.g., if the integrand is $$(x^2+a^2)^{-\frac32}$$ then on making this substitution we find

 * $$\begin{align}

\int\limits_0^z(x^2+a^2)^{-\frac32}dx&=a^{-2}\int\limits_0^\alpha \cos(\theta)d\theta&z>0\\ &=a^{-2}\Big[\sin(\theta)\Big]_0^\alpha&\alpha=\arctan\left(\tfrac{z}{a}\right)\\ &=a^{-2}\sin(\alpha)\\ &=a^{-2}\frac{\frac{z}{a}}{\sqrt{1+\frac{z^2}{a^2}}}\\ &=\frac{z}{a^2\sqrt{a^2+z^2}} \end{align}$$

If the integral is

 * $$I=\int\limits_0^z\sqrt{x^2+a^2}\qquad z>0$$

then on making this substitution we find

 * $$\begin{matrix}

I & = & a^2 \int_0^\alpha \sec^3 \theta \, d\theta & & & \alpha = \tan^{-1} (z/a) \\ & = & a^2 \int_0^\alpha \sec \theta \, d\tan \theta & & & \\ & = & a^2 [ \sec \theta \tan \theta ]_0^\alpha & - & a^2 \int_0^\alpha \sec \theta \tan^2 \theta \, d\theta & \\ & = & a^2 \sec \alpha \tan \alpha & - & a^2 \int_0^\alpha \sec^3 \theta \, d\theta & + a^2 \int_0^\alpha \sec \theta \, d\theta \\ & = & a^2 \sec \alpha \tan \alpha & - & I & + a^2 \int_0^\alpha \sec \theta \, d\theta \\ \end{matrix}$$

After integrating by parts, and using trigonometric identities, we've ended up with an expression involving the original integral. In cases like this we must now rearrange the equation so that the original integral is on one side only

 * $$I$$
 * $$=\tfrac{a^2}{2}\Bigg(\sec(\alpha)\tan(\alpha)+\int\limits_0^\alpha \sec(\theta)d\theta\Bigg)$$
 * $$=\tfrac{a^2}{2}\Bigg(\sec(\alpha)\tan(\alpha)+\bigg[\ln\Big(\sec(\theta)+\tan(\theta)\Big)\bigg]_0^\alpha\Bigg)$$
 * $$=\tfrac{a^2}{2}\bigg(\sec(\alpha)\tan(\alpha)+a^2\ln\Big(\sec(\alpha)+\tan(\alpha)\Big)\bigg)$$
 * $$=\tfrac{a^2}{2}\bigg(\tfrac{z}{a^2}\sqrt{a^2+z^2}+\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * $$=\tfrac{1}{2}\bigg(z\sqrt{z^2+a^2}+a^2\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * }
 * $$=\tfrac{a^2}{2}\bigg(\sec(\alpha)\tan(\alpha)+a^2\ln\Big(\sec(\alpha)+\tan(\alpha)\Big)\bigg)$$
 * $$=\tfrac{a^2}{2}\bigg(\tfrac{z}{a^2}\sqrt{a^2+z^2}+\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * $$=\tfrac{1}{2}\bigg(z\sqrt{z^2+a^2}+a^2\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * }
 * $$=\tfrac{1}{2}\bigg(z\sqrt{z^2+a^2}+a^2\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * }
 * $$=\tfrac{1}{2}\bigg(z\sqrt{z^2+a^2}+a^2\ln\left(\tfrac{z+\sqrt{a^2+z^2}}{a}\right)\bigg)$$
 * }

As we would expect from the integrand, this is approximately $$\frac{z^2}{2}$$ for large $$z$$.

In some cases it is possible to do trigonometric substitution in cases when there is no $$\sqrt{\ \ }$$ appearing in the integral.

Example
 * $$\int\frac{dx}{x^2+1}$$

The denominator of this function is equal to $$\bigl(\sqrt{1+x^2}\bigr)^2$$. This suggests that we try to substitute $$x=\tan(u)$$ and use the identity $$1+\tan^2(u)=\sec^2(u)$$. With this substitution, we obtain that $$dx=\sec^2(u)du$$ and thus
 * $$\int\frac{dx}{x^2+1}=\int\frac{\sec^2(u)}{\tan^2(u)+1}du$$

 * $$=\int\frac{\sec^2(u)}{\sec^2(u)}du$$
 * $$=\int du$$
 * $$=u+c$$

Using the initial substitution $$u=\arctan(x)$$ gives
 * $$\int\frac{dx}{x^2+1}=\arctan(x)+C$$

Secant substitution

If the integrand contains a factor of the form $$\displaystyle\sqrt{x^2-a^2}$$ we use the substitution
 * $$x=a\sec(\theta)\quad dx=a\sec(\theta)\tan(\theta)d\theta\quad\sqrt{x^2-a^2}=a\tan(\theta)$$

Example 1
Find $$\int\limits_0^z \frac{\sqrt{x^2-1}}{x}dx$$.

 * $$\int\limits_0^z \frac{\sqrt{x^2-1}}{x}dx$$
 * $$=\int\limits_0^\alpha \frac{\tan(\theta)}{\sec(\theta)}\sec(\theta)\tan(\theta)d\theta\qquad z>1$$
 * $$=\int\limits_0^\alpha \tan^2(\theta)d\theta\qquad\qquad\qquad\qquad\alpha=\arcsec(z)$$
 * $$=\bigg[\tan(\theta)-\theta\bigg]_0^\alpha\qquad\qquad\qquad\tan(\alpha)=\sqrt{\sec^2(\alpha)-1}$$
 * $$=\tan(\alpha)-\alpha\qquad\qquad\qquad\qquad\tan(\alpha)=\sqrt{z^2-1}$$
 * $$=\sqrt{z^2-1}-\arcsec(z)$$
 * }
 * $$=\bigg[\tan(\theta)-\theta\bigg]_0^\alpha\qquad\qquad\qquad\tan(\alpha)=\sqrt{\sec^2(\alpha)-1}$$
 * $$=\tan(\alpha)-\alpha\qquad\qquad\qquad\qquad\tan(\alpha)=\sqrt{z^2-1}$$
 * $$=\sqrt{z^2-1}-\arcsec(z)$$
 * }
 * $$=\sqrt{z^2-1}-\arcsec(z)$$
 * }
 * $$=\sqrt{z^2-1}-\arcsec(z)$$
 * }

Example 2
Find $$\int\limits_1^z \frac{\sqrt{x^2-1}}{x^2}dx$$.

 * $$\int\limits_1^z \frac{\sqrt{x^2-1}}{x^2}dx$$
 * $$=\int\limits_1^\alpha \frac{\tan(\theta)}{\sec^2(\theta)}\sec(\theta)\tan(\theta)d\theta\qquad z>1$$
 * $$=\int\limits_0^\alpha \frac{\sin^2(\theta)}{\cos(\theta)}d\theta\qquad\qquad\qquad\alpha=\arcsec(z)$$
 * }
 * $$=\int\limits_0^\alpha \frac{\sin^2(\theta)}{\cos(\theta)}d\theta\qquad\qquad\qquad\alpha=\arcsec(z)$$
 * }

We can now integrate by parts

 * $$\int\limits_1^z \frac{\sqrt{x^2-1}}{x^2}dx$$
 * $$=-\bigg[\tan(\theta)\cos(\theta)\bigg]_0^\alpha+\int\limits_0^\alpha \sec(\theta)d\theta$$
 * $$=-\sin(\alpha)+\bigg[\ln\bigl(\sec(\theta)+ \tan(\theta)\bigr)\bigg]_0^\alpha$$
 * $$=\ln\bigl(\sec(\alpha)+\tan(\alpha)\bigr)-\sin(\alpha)$$
 * $$=\ln\bigl(z+\sqrt{z^2-1}\bigr)-\frac{\sqrt{z^2-1}}{z}$$
 * }
 * $$=\ln\bigl(\sec(\alpha)+\tan(\alpha)\bigr)-\sin(\alpha)$$
 * $$=\ln\bigl(z+\sqrt{z^2-1}\bigr)-\frac{\sqrt{z^2-1}}{z}$$
 * }
 * $$=\ln\bigl(z+\sqrt{z^2-1}\bigr)-\frac{\sqrt{z^2-1}}{z}$$
 * }
 * }

Exercise
Evaluate the following using an appropriate trigonometric substitution.


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Engineering Acoustics/Transverse vibrations of strings

Introduction
This section deals with the wave nature of vibrations constrained to one dimension. Examples of this type of wave motion are found in objects such a pipes and tubes with a small diameter (no transverse motion of fluid) or in a string stretched on a musical instrument.

Stretched strings can be used to produce sound (e.g. music instruments like guitars). The stretched string constitutes a mechanical system that will be studied in this chapter. Later, the characteristics of this system will be used to help to understand by analogies acoustical systems.

What is a wave equation?
There are various types of waves (i.e. electromagnetic, mechanical, etc.) that act all around us. It is important to use wave equations to describe the time-space behavior of the variables of interest in such waves. Wave equations solve the fundamental equations of motion in a way that eliminates all variables but one. Waves can propagate longitudinal or parallel to the propagation direction or perpendicular (transverse) to the direction of propagation. To visualize the motion of such waves click here (Acoustics animations provided by Dr. Dan Russell,Kettering University)

One dimensional Case
Assumptions :

- the string is uniform in size and density

- stiffness of string is negligible for small deformations

- effects of gravity neglected

- no dissipative forces like frictions

- string deforms in a plane

- motion of the string can be described by using one single spatial coordinate

Spatial representation of the string in vibration:

The following is the free-body diagram of a string in motion in a spatial coordinate system:

From the diagram above, it can be observed that the tensions in each side of the string will be the same as follows:

Using Taylor series to expand we obtain:

Characterization of the mechanical system
A one dimensional wave can be described by the following equation (called the wave equation):

$$\left( \frac{\partial^2 y}{\partial x^2} \right)=\left( \frac{1}{c^2} \right)\left( \frac{\partial^2 y}{\partial t^2} \right)$$

where,

$$y(x,t)= f(\xi)+g(\eta)\,$$ is a solution,

With $$\xi=ct-x\,$$ and $$\eta=ct+x\,$$

This is the D'Alambert solution, for more information see:

Another way to solve this equation is the Method of separation of variables. This is useful for modal analysis. This assumes the solution is of the form:

$$ y(x,t)= f(x)g(t)\ $$

The result is the same as above, but in a form that is more convenient for modal analysis.

For more information on this approach see: Eric W. Weisstein et al. "Separation of Variables." From MathWorld—A Wolfram Web Resource.

Please see Wave Properties for information on variable c, along with other important properties.

For more information on wave equations see: Eric W. Weisstein. "Wave Equation." From MathWorld—A Wolfram Web Resource.

Example with the function $$f(\xi)$$ :

Example: Java String simulation

This show a simple simulation of a plucked string with fixed ends.

Back to Main page


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Blended Learning in K-12/Blended Learning In Grades K-2

Blended learning within early childhood education is an interesting concept; how can a student who lacks the ability to read and write be part of a virtual community? While many of the children at the primary level lack the ability to read, incorporating technology enhanced learning is still a reality. Teachers can enhance an existing curriculum, improve communication with the school community and devise forums which reinforce & enrich the early childhood education. Even though adaptations must occur in order for e-learning to be successful with young children, primary students "should not be excluded from the virtual learning world simply because of their age and developmental levels" (Scott; 2003).

When the topic of blended learning arises people often think of students meeting within a classroom setting and then continue the learning experience online in the comfort of their home. However within primary classrooms, blended learning can be more comparable to technology integration; serving the class environment as a teaching aide. Since many primary classrooms now have a technology center which can include anywhere from one to half a dozen computers e-learning is becoming a reality. While this is not the true definition of blended learning, this type of face-to-face instruction followed by independent activities based on individual student needs is the building blocks for higher level blended learning.

Blended Education: Application Examples

Curriculum

An overview of the primary classroom sees children learning to read, beginning to add, and exploring numerous topics for the first time. Most classrooms are brimming with children, lacking an aide and overloaded with information. By investigating each subject within a primary classroom, teachers can envision how blended learning can be a real part of early childhood education.

Incorporation of technology into the primary classroom can be as a simple as bringing the students to a website which better illustrates a story explored in class. For example, if a class reads "Brown Bear, Brown Bear, What Do You See?" written by Bill Martin, Jr. to further extend upon the story a primary teacher may set a website such as Animal Vocabulary on a computer in the technology center.

Language arts within a primary classroom can be enhanced with a learning community such as The Monster Exchange. This website has the children collaborating with children all over the United States and beyond to work on descriptive writing. Children draw a picture of a monster and write a description of the creature in class, the teacher inputs the picture and description into the website. Classrooms connect with another class, read one anothers' descriptions, and then try to recreate the original monster. Pictures are set up side by side in the Monster Gallery for comparison. Children can access the monster website at home and work collaborately with their families.

A primary teacher knows all too well that purchasing enough math manipulatives for the whole class can be quite expensive and often not a reality for every school. Technology and blended learning offer a solution. If an educator works with his/her class on a lesson about patterns, he/she can direct students to practice the lesson on Virtual Library of Math Manipulatives. To begin this lesson the teacher would explore patterns in numerous ways using varying sets of manipulatives to illustrate patterns; however when it comes to having a set of coins or buttons for each student this might not be a feasible. Solution: if a teacher uses this website within a lab setting in which all the students are using a computer, children can play with patterns independently, make mistakes, ask questions and the best part no buttons, coins or colored bears to clean up.

Often time history is neglected because the basic skills often take top priority in the primary classroom. Using e-learning is a great way to further explore topics such as history. One very interactive website explores George Washington. During the month a February a 3rd grade teacher, for example, can set this website up as a favorite within her computer center. Each child can be required to view the site and make a comment in the notebook next to the computer. Those comments can later be shared as a class.

Since science resources are so abundant online, we can look at blended learning from a different angle. A second grade class designed this Space Website. After learning about space in class, the children worked within groups to develop a virtual learning area made for children in primary grades. The students and teacher took what they knew, blended it into technology and now other students can benefit.

Communication

Blended communication could be the most successful form for the new generation of parents. Quite often information relayed to a primary student quite often does not make it to the ears of a parent. Besides traditional classroom visits, parent/teacher conferences and telephone calls, many teachers of all students are realizing that reaching parents through emails, websites and discussion boards are more fruitful in contacting parents. Designing an online community where teachers can post and explain information about their teaching methods can help clarify classroom procedures. With the same regard, parents can ask questions, review announcements, and become an active part of the classroom through a virtual environment. Searching within Yahoo groups, numerous groups can be discovered which join parents and education groups. “Some schools are exploring the use of video conferencing and 'streamed' (stored for viewing at home) videos to promote parent understanding and involvement in student learning" (Starr, 2005).

This blended communication is even opening up a place for parent input to class learning. Teachers can design questions through online questionnaires from places like SurveyKey. Educators can ask parents about issues with in the class, specific needs and concerns. As parents respond, a teacher can make adjusts and improvements. Once again this is extremely important within younger students, as they often have a difficult time expressing experiences which they may have in class.

Reinforcement & Enrichment

Teachers at every level grapple with the difficulty of addressing the needs of each child within a classroom, however this challenge is extremely prevalent within the early childhood classroom as students are exploring the building blocks of education. This challenge can be aided with blended learning.

Studies have been surfacing for years that foreign language instruction should begin at the elementary level instead of postponing that learning until high school, however due to budgetary concerns, foreign language classes seem like a frill (Walker, 2004). By teaching another language to young children, we give them the greatest chance to fully absorb a second language. If an elementary school does not offer a foreign language classes, teachers and parents can still expose primary students to another language through technology. From simple websites which vocalize the French Alphabet to websites which allow the students to progress through activities to learn Spanish.

Within primary grades a child many times needs extra practice. The web has the amazing ability to give kids extra help in a way different from group classroom instruction, maybe in a form in which a child learns better. For example, if a teacher has introduced new letter sounds and she/he notices a student is struggling, the student can either use the computer center to practice or a Phonics website address can be send home for parents to use as practice.

Adaptations for Blended Learning in Early Childhood Education

One major concern for young children on the Internet is safety. While students within upper grades understand the seriousness of broadcasting their personal information, often younger students are ignorant to that fact. In order to protect the identity of students, it is recommended that students work as a group. Since group work is quite prevalent within a primary classroom, it is very realistic that young children work as a group within a virtual learning environment, collaborating on answers to contribute. Students can work as a face-to-face group in the classroom, develop an answer and post the response within their virtual environment. Working as a group also alleviates the need to post responses using full names, pictures and other personal information, instead the children post as, for example, The Green Group.

The design of a virtual community needs to be adapted for the younger set. Since the literacy development varies greatly within grades K-2, sites should use pictures and common shapes to navigate through the information. As streaming video and digital voice technologies improve and are becoming more common site participation becomes more user friendly for those with limited reading skills. Responses within an online learning environment need to be configured with developmental needs in mind. Answers may need to be multiple choice or give the contributor the ability to "draw" an answer.

Sites which are made specifically for online classes such as Moodle or other course management systems are not appropriate for the early childhood environment because "younger students may not have the study skills, reading abilities and self-discipline to fare well without a class to go to"(Russo, 2001). That does not mean they need to be excluded from the virtual community, we just need to think of these years as their preparation for becoming part of a Brave New World of teaching.


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Blended Learning in K-12/Blended Learning in Grades 3-6

Starting Out With Blended Learning

As students become more confident of their technology skills in grades 3-6, and access to technology at home increases, the opportunities for blended learning experiences broaden. Web-based resources can provide more indepth information on academic topics, support slow learners, and enrich high achievers. Communication between home and school can be vastly improved by utilizing the web to improve the learning experience.

One simple way to begin using technology with students in 3rd through 6th grade is through an asynchronous communication such as the use of e-mail. There are numerous ways in which e-mail can be helpful in the classroom setting. Using e-mail as part of a blended learning experience can enhance a face-to-face discussion and allow students to further explore their learning. Many students already have a home e-mail account which they use to communicate with their friends or family, and by the age of 10, students are mature enough to learn how to use email. Teachers can make themselves available to students and parents through email, to answer questions on specific topics and to discuss classroom topics. Students can stay in touch with the teacher if they are absent.

Another popular use of e-mail is keypals, in which students are matched up with students from other schools and participate in an exchange of information and ideas. Keypals help the students see themselves as part of worldwide learning community, and learn about other cultures and ways of life. Another use of email is to adopt a grandparent. More and more senior citizens are becoming technology savvy, and would love to exchange information with students. There are some safety measures that teachers need to set forth if adopting a grandparent. Full student names, addresses, and phone numbers should not be given out at any time. This is to ensure the safety of the students in today's world.

Instant Message (IM) is a synchronous form of communication that can be started in the middle grades. IM allows instantaneous feedback from teachers or students. Students can ask questions of other students or the teacher about an assignment or participate in a discussion with a teacher or classmate about concepts or topics being covered in class. IM does have some drawbacks. Students can send inappropriate messages or pictures to other students or teachers, so it is important that students are instructed on acceptable use. Restricted accounts, which many parents use for their children, might block the use of instant messaging.

Improving Home-School Communication

Getting parents involved in their child's education is key to academic success. Teachers can publish web pages linked to the school website to provide a multitude of information for parents. Teachers can provide a weekly agenda of what's going on in class, they can provide detail of homework assignments, permission slips for field trips, and much more, to be available to parents trying to stay on top of their child's education. Many teachers use blogs for this purpose, which might be a simpler way to get information online immediately. Teachers can also link to websites that enhance or expand on topics covered in class.

There is no doubt that technology can improve parent-teacher communication. Through the use of Edline or classroom websites, parents can stay more involved in what their student is doing in the class and also how they are doing in the class. If a parent can quickly view what their child has to do or see an area where they need assistance, it can make for easy communication with the teacher about what needs to be done. For success to be evident, there must be good communication between the parents and the teachers. --Nick Hartz

Curriculum Connections

Many online activities are available for 3rd through 6th grade students that provide extra practice on classroom topics, or expand and enrich learning. Teachers can link to these sites so they are available to students outside of school. Across the curriculum the web offers resources that engage students in the learning process, and will actually make them want to spend additional time outside of school on learning.

It is important when planning a blended learning lesson within the primary grades to focus on a unit of study, then intertwine it with technology. The educators at San Diego State University have designed a tool to aide teachers in their preparation.

For teachers looking to integrate science experimentation into their middle grades' curriculum, a wonderful interactive site is Zoom Kitchen Chemistry. Here, students can conduct virtual experiments to learn about real-life chemical reactions, or find out about real science experiments they can do at home with items found in their own kitchens. This site is wonderful for extending classroom learning using technology. If the class is studying space and the solar system, an excellent resource for young astronomers is Star Child. Here students can find information about space topics, utilize simulations and a glossary of space terms.

One method of blended learning in math is to have students practice their math facts online. This provides the opportunity for students to spend extra time practicing if needed. At Math Magician students can have fun interactively while working on math facts, all operations, and two levels are available, so more advanced students can progress at their own pace through more challenging material. If the teachers seeks ways of using manipulatives to teach math, a wonderful site that utilizes java applets so students can have a hands-on experience is The National Library of Virtual Manipulatives.

A great source for students to work on their reading skills is The Reading Matrix. There are numerous reading activities ranging from vocabulary, comprehension, to proofreading, and short stories. Many of these sites provide an online quiz for the students to take. Teachers can find good sites on this page by looking at the ratings that it has received.

Students can have a blast at National Geographic for Kids. Students will spend hours going through the website which contains, quizzes, games, cartoons, and excellent information. This site is great for any social studies buff, or anyone that wants to have fun while doing research on the web. This site will have students talking about social studies for the entire quarter.

Everyone wants to create their own music that they can listen to. The website Creating Music (which requires Java) allows students to create their own musical sketch pad and then listen to what they have designed. Students can learn about beat, tempo, and rhythm while enjoying being the composer of music. This is a great site for elementary students to learn about music and to get them interested without having to pick up a single instrument.

Virtual Field Trips Field trips are a large part of any classroom. Quite often a teacher would love to take her students places to which a bus trip is not an option. Technology offers the solution with virtual field trips. Students can look at museum artifacts, visit an aquarium, or admire beautiful art while sitting with their class. Sharing a field trip virtually is also a great way to reflect a on a trip and share experiences with future classes. For example, each year students from Bennet School in North Carolina design a website about their trip to the State Capital, each year the website gets remodeled, but old versions are kept online to serve as a scrapbook.

Video Conferencing

How about a field trip without even leaving the room? With the creation of video conferencing this is made possible to all students and teachers to further enhance their students learning and enthusiasm. Students love to take field trips and they love to go on the computer, now teachers can have the best of both worlds. Here are some sites that encompass a virtual field trip.

Science Center is a great site for science educators who want to have their students learn first hand about the human body, space, dinosaurs, and eyes. For this field trip a fee of $150.00 is required for a 45-minute tour. This might not be feasible for schools who are on a very small budget. They do provide a 25 minute project for about $100.00.

Ever want to receive video feeds from underwater? The Aquatic Research Interactive Site does just that. Science teachers can have students watch video streams from underwater for numerous topics. This site has been designed for teachers and students to better understand concepts below the Earth's surface. Not a science teacher? Math, history, and physics teachers can also benefit from this site as well. There is one major drawback to this site, the fee. A whopping $195.00 fee for the use of the video clips are required.

Teacher's Pet? That might be what students will be talking about after a trip to The Bronx Zoo. This two way interactive site is designed for the elementary students to learn more about an animal's behavior. How about having a lion in your class? This site allows a class to have them and you don't have to worry about students with allergies or a student being attacked. This site is sure to have your students talking for a long while. This fieldtrip is $125.00 for a maximum for 35 students.

Conclusion: Can blended learning work in grades 3-6? That depends on the teacher, tech support, students, and administration. Lisa Abate understands that a blended learning classroom will require more work. She mentioned that she spent a lot of her time "troublehooting studet problems (such as lost passwords)." (Abate, 2004) Essentially, doesn't education come down to "are students learning?" If students log on to a website long after they are finished woith their assignment to further enrich their studies have teachers accomplished their goals? Is the time worth the satisfaction a student gets by learning more than have to? In Lisa's beta test of her math classroom she found that students were spending more time than needed on specific activities. (Abate, 2004) This is a teacher's dream come true, students spending more time than is required on assignments.


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Blended Learning in K-12/Blended Learning In Grades 7-8

Blended Learning Grades 7–8
As students mature and can handle learning without constant teacher attention, online applications may become more effective for teaching some curriculum. After teachers and students feel comfortable with e-mail and webpage design, they can dig further into the realm of blended learning by accessing some excellent websites. All of these sites are designed to help the learner better understand a topic. These are just a few good examples of websites; there are numerous other sites available.

Triple A Math This site is great for K-8 math teachers because of its content. Students can read the explanation of each measurement, play some challenge games, and some interactive practices. Students will have endless hours of fun checking out this site.

Seattle Art Museum (Science, Social Studies) This site is for the "explorer" learner. Students can learn about navigation techniques used in the Age of Exploration. The site includes a video clip of how to use an octant. This was very cool to see in addition to learning about the navigation used today.

Hands On the Land Imagine collecting water samples, monitoring the ground ozone levels, and more on this environmental website. The site provides interaction with other schools, students, and the forest preserve as students are engaged while learning about the environment. There are even lesson plans for teachers to further enhance the students learning.

BioPoint This is an online site for teacher created webquests. They are listed by grade level and have everything you need to deliver a unit.

Barking Spiders Poetry Barking Spiders is a site devoted to poems for children. Students can design their own poems online by filling in the blanks. There are mazes that can be completed and poems to be read.

 The Smithsonian is known world-wide for its magnificent collections. Since not every class can take a trip to these free museums, the Smithsonian has made it easy for every class to access the collections and activities to go with them online. Teachers can search for lessons by state standard quickly and easily. Teachers can also check out resources that can be sent to their classrooms.

Social Science Some teachers are more comfortable with enhancing their class through projects. A sample of one would be a project developed by Kate Purl at Urbana Middle School. The year was ended with a seventh grade project on Africa. Each student had to research an area of Africa, learning specifics about the area such as flora and fauna. The information they found was then added to a website. Once all the areas of Africa were added, the adventure began. As teams, the students had to traverse the African continent. The information that the students provided was combined into a webquest that was a toss up between "Where in the World is Carmen San Diego?" and "The Oregon Trail." Decisions made as you traveled the trail could lead to success or failure as an African explorer. The site is still available at: Africa: Choose Your Own Adventure.

Other successful online adventures for Middle School students might be Virtual Ancient Civilizations which is a work in progress:

Numerous Subjects Quiz Hub. What can't you do at this site? The site is loaded with news quizzes, online maps, chess games, concentration games and much more. Your students will never want to leave the site. This will have them engaged for hours upon hours. There is even an art area where you can design whatever you want. This was my favorite link!!

Benefits for Students Research shows that students who are involved in online learning during the middle school years are more likely to keep their academic grades higher than those who are not exposed to online learning (Belanger, 2005). Additionally, the attendance rate of students using computers is higher along with the ability of students to do well in group situations and within project-based instruction. Blended learning is also research based in that it pulls from research done by Piaget, Vygotsky, Bloom Keller and Gery. Using online education for middle schools students is a viable way of enhancing the curriculum by providing live events such as online homework help. This could come from the teacher, or it could be from other students in a cooperative learning environment. Blended learning also gives the students a chance to move at their own pace. What one student may be able to learn quickly another may need more time to digest and understand. In blended learning, students moves at their own pace and they have the availability of a teacher when questions arise. There is also a collaborative element to blended learning. If a student does not understand part of an assignment, there are other students available to ask questions. Students can use a chat room, IM or e-mail to work together to complete a project. Along with the curriculum, testing can also take place online. This would allow students to take assessments when they feel they are ready. It would also allow for quick feedback from those assessments so that students know where they stand with respect to their grade. Lastly, when students use the web for learning there is a wealth of materials available to them at the click of a mouse. Dictionaries, encyclopedias and research are just some of the information that students can access during their learning experience. These five ingredients to blended learning are important for the students to have the best possible educational experience. It allows them to remain anonymous so that their mistakes are not broadcast throughout the classroom, giving them self confidence. It is a way to allow students to take control of their own learning experience.

The Down Side As with every advance in technology, blended learning can have its faults also. While online courses have been available for several years to the post secondary student, use of online technology is generally thought of as an enhancement for the secondary schools. Schools are using computers more than ever before with an increase from 60% in 1993 to 84% in 2001 and home use of computers growing during the same time from 25% of students to 66% (NCES, pg 1). Schools might use computers more if computers were available in each classroom. Many schools don't have that luxury and, instead, must move students to a central computer lab to make use of the technology available. This creates more work for teachers and takes away from instructional time. Most will also admit that it is the teacher, and her comfort with technology, that affects how well that technology is presented in the classroom. In his book, “Oversold and Underused,” Larry Cuban states that computers are not successful because teachers who use computers for instruction do so infrequently and unimaginatively (Harvard University Press, 2001).

Taking Technology Further Some students, for various reasons, need to work at their own pace. While distance learning got its start at the post-secondary level, it is slowly gaining momentum at the middle and high school level. Students who are gifted, challenged or have health reasons and do not do well in a traditional learning environment now have the opportunity to complete their education online. Several schools are now in operation that offer aid to these students. Some worth looking at are:

Advanced Academics

Colorado

Exel High School High School

James Madison High School Online

Teacher Controlled Blended Learning As with any level of education, there are some teachers who prefer to have control over what their students are doing online. Instead of accessing one of the above websites or a WebQuest, these teachers choose to develop their own curriculum for their students. This curriculum needs to be well thought out before anything is ever put online. New South Wales Department of Education and Training has some guiding questions that teachers should ask themselves before they begin to design a web-enhanced course. They first suggest that you know WHY you are trying to place information online for your students. Once you are sure that what you are looking for is not already online, then it is time to understand the goals that you have for you students regarding the online content: Do you want them to learn to work independently? Or do you want to free up your time to work with those students who need the extra help? The other suggestions include teaching your students slowly, showing them step by step what you expect from them. However, the ultimate suggestion was that if a teacher were to be interested in providing web-enhanced learning for their class, they should have had the experience of online learning themselves (NSWDE, 2005).