Tuesday, October 28, 2008
In every high school textbook, the trapezoid in included as one of the quadrilaterals to study (or investigate). In almost every one, it is defined as a quadrilateral with exactly one pair of parallel sides. In this talk, I will argue that the definition should be changed and that there is much more to the trapezoid than is given in the books.
Almost every theorem about a trapezoid can be broken into two categories – those that are really about parallel sides and those that in some way incorporate a feature of a quadrilateral.
In the history of geometry, many definitions have been changed, some from being exclusive to being inclusive. An exclusive definition is one that separates other objects from being in the same class. For example, in Euclid, an isosceles triangle is defined to be a triangle with exactly two equal sides. Thus, an equilateral triangle is not isosceles. Likewise all his definitions of quadrilaterals are exclusive. During the next 2300 years, most of these have been changed. Now in every high school textbook, equilateral triangles are isosceles, rectangles and rhombi are parallelograms, and squares are rectangles and rhombi. There are two advantages to having inclusive definitions – (1) theorems for the more restricted case become corollaries for the more general case and (2) converses do not need to contain an “or” conclusion.
By one construction, we can easily see that the parallelogram is a special case of a trapezoid. Take two points on each of two parallel lines. The quadrilateral formed by using these points is a trapezoid (or a parallelogram). Consequently, the definition can be strengthened by including it as such. So why is the definition maintained in the textbooks? I think primarily that many authors do not wish to go against standard terminology, most of the authors have not thought about the inconsistency in terminology, they are not actively engaged in discovering and proving theorems in geometry, and there are no converses for the trapezoid covered in high school geometry.
Thus, for good mathematical reasons let’s change the definition to:
A trapezoid is a quadrilateral with at least one pair of parallel sides.
In the theorems that follow, some require that a pair of sides be non-parallel, but the parallel case follows as well, usually with little or no additional proof.
While we are at this change, the same argument applies to the isosceles trapezoid and the rectangle. A simple construction shows that the rectangle is a special case of the isosceles trapezoid. How then can we define the isosceles trapezoid so that the rectangle is a special case. I offer a variety of different definitions.
(1) An isosceles trapezoid is a cyclic trapezoid.
(2) An isosceles trapezoid is a trapezoid with a pair of supplementary opposite angles.
(3) An isosceles trapezoid is a trapezoid with the other pair of sides anti-parallel with respect to the parallel sides.
(4) An isosceles trapezoid is a trapezoid with a pair of congruent base angles.
(5) An isosceles trapezoid is a trapezoid with congruent diagonals.
Here are a few other definitions that I offer.
The diacenter of a quadrilateral is the intersection point of the diagonals.
A quord of a quadrilateral is a segment with endpoints on two sides of a quadrilateral.
A median is a quord with endpoints the midpoints on opposite sides.
Thursday, October 16, 2008
Thursday, October 2, 2008
Let me explain this through an example:
Needless to say, CAT preparations will bring you one of the most memorable times of your life. Even with all that stress, those class exercises, those depressing mocks and the fierce competition, CAT preparation is so much fun.
Given below are some teaser questions for you to try your hands on. Good luck!
Sunday, September 28, 2008
Psychology Articles by Uncommon Knowledge (Psychology Articles by Uncommon Knowledge )
Recent Psychology Articles (Results 1 - 15 of about 453) (Recent Psychology Articles (Results 1 - 15 of about 507) )
The 50 Most-Frequently Read Articles in Sociology (The 50 Most-Frequently Read Articles in Sociology )
sociology - List of sociology topics (sociology - List of sociology topics )
Literary Genres and Learning How to Write (Literary Genres and Learning How to Write )
Arts & Literature Articles (Arts & Literature Articles )
TheHistoryNet: From the World's Largest History Magazine Publisher (TheHistoryNet: From the World's Largest History Magazine Publisher )
Political and Religious Events in North America and Europe (Political and Religious Events in North America and Europe )
Magazine Articles on International Politics (current) (Magazine Articles on International Politics (current) )
Government & Political Articles (Government & Political Articles )
Geography - Articles (Geography - Articles)
EZGeography - Hundreds of Geography Articles ( EZGeography - Hundreds of Geography Articles)
Economics - Articles ( Economics - Articles)
Europe Economics: Articles and Working Papers ( Europe Economics: Articles and Working Papers)
Magazine Articles on Strategy & Management (current) ( Magazine Articles on Strategy & Management (current))
Management Articles (Management Articles )
Management Methods | Management Models | Management Theories
Category:Management - Wikipedia, the free encyclopedia
Business Management, Leadership and Decision Making Strategies and Resources - BetterManagement.com
Moneyterms: investment/finance glossary/dictionary/education
BusinessDictionary.com - Online Business Dictionary
My experiences I feel nostalgic
guys while tackling with RCs please keep the following points in mind ...
Step 1) Try to get an estimate of your reading speed and comfort level with various topics given in CAT first (You can try your hand @ How to read faster & better by Norman Lewis)
Step 2) Read on diverse topics regularly. Summarize whatever you have read and note down the important points. (would be helpful for GD)
Step 3) Practice a lot from the previous years mock papers
I am attaching some material which might be helpful
You can refer to the following links as well
2. English Learning for Beginners ESL EFL learning materials including grammar, listening, vocabulary, reading, writing, reference, quizzes and an online course, as well as free lesson plans for ESL EFL Teachers.
3. Capital Community College - Redirect Page
FirstScience - Science Articles
Science and technology information from Scientific American
Science Daily: News & Articles in Science, Health, Environment & Technology
Philosophy & Religious Articles
Philosopher Biographies and Ideas
Psychology Articles by Uncommon Knowledge
Recent Psychology Articles (Results 1 - 15 of about 507)
The 50 Most-Frequently Read Articles in Sociology
sociology - List of sociology topics
Literary Genres and Learning How to Write
Arts & Literature Articles
TheHistoryNet: From the World's Largest History Magazine Publisher
Political and Religious Events in North America and Europe
Magazine Articles on International Politics (current)
Government & Political Articles
Geography - Articles
EZGeography - Hundreds of Geography Articles
Economics - Articles
Europe Economics: Articles and Working Papers
Magazine Articles on Strategy & Management (current)
FirstScience - Articles (FirstScience - Science Articles)
Science and technology information from Scientific American (Science and technology information from Scientific American)
Science Daily: News & Articles in Science, Health, Environment & Technology ( Science Daily: News & Articles in Science, Health, Environment & Technology)
Philosophy & Religious Articles (Philosophy & Religious Articles)
religion online (religion online )
Philosopher Biographies and Ideas (Philosopher Biographies and Ideas)
Articles (http://www.prs-ltsn.leeds.ac.uk/philosophy/articles/index.html )
** Also refer to The Guardian for some nice articles.
1. Try to read the whole text of the passage once, if possible. Many people think you should just skim the passage or read the first lines of every paragraph, and not to read the passage. We believe this is an error: if you misunderstand the main idea of the passage, you will certainly get at least some of the questions wrong. Give the passage one good read, taking no more than 3 minutes to read all of the text. Do not read the passage more than once – that wastes too much time. If you have not understood it completely, try to answer the questions anyway. Note: this point of reading the whole passage is important for test-takers whose first language is not English, provided that they can read the passage in 3 minutes or less.
2. Make brief notes on the text on your scrap paper. As we will see below in greater detail, you should write down a couple of words on A) the Main Idea or Primary Purpose, B) Organization/Structure of the passage, and C) the Tone or Attitude of the author (if applicable). You just need a few words for each of these areas, and altogether it should not take longer than 30 seconds to write down.
3. Remember that the tone or attitude of the passage is usually respectful and moderate, never going to extremes of praise nor criticism. ETS obtains its Reading Comprehension passages from real articles about real academics and professionals. So the tone of the articles, even when there is criticism in the passage toward an academic or her work, is always balanced and moderate. In the same vein, articles that deal with minorities or ethnic groups are almost always positive and sympathetic.
4. Look out for structural words that tell you the important ideas or transitions in a passage.
Continue the Idea Words
In the same way
Contradiction or Contrast Words
In spite of
On the one hand…on the other hand
5. Go back to the text of the passage for the answers. Many test-takers fail to return to the text of the passage to look for the correct answers. They rely solely on their memories and understanding of the passage after having read or skimmed it. Wrong. ETS is counting on that. Go back to the text to look for information to answer the questions. Nine times out of ten, the answer lies within the passage.
Of the 6 most important types of questions for Reading Comprehension, we will first look at Main Idea/Primary Purpose Questions, and the strategies we can use to answer them.
Main Idea/Primary Purpose Questions
Many people believe there is no difference between the main or central idea of the passage and the primary purpose of the author of the passage. This is simply not true. Let's take a look at the subtle but important difference between them:
The question might look something like this:
"Which of the following best states the central idea of the passage?"
"Which of the following most accurately states the main idea of the passage?"
"Which of the following is the principal topic of the passage?"
"The main topic of the passage is...."
The question might look like this:
"The primary purpose of this passage is to..."
"The primary purpose of the passage as a whole is to..."
"The primary focus of this passage is on which of the following?"
"The main concern of the passage is to..."
"In the passage, the author is primarily interested in...."
"The passage is chiefly concerned with..."
Main Idea: Look in the first and last paragraphs for the main idea. Any conclusion words like therefore, thus, so, hence, etc. that you see are most likely introducing the main idea. The correct answer will say the same thing as it says in the text, but using different words. The Main Idea is not always stated explicitly in the passage – in fact, more likely than not, it is not stated explicitly. Therefore, in order to answer this type of question when it is more implicit:
1. Re-read the first line of every passage, and the last line of the first and last paragraphs. This should give you the general structure or outline of the argument, with which you can answer the Main Idea question.
2. After determining the general structure or content of the argument, eliminate answer choices that are too broad or too specific, i.e. answer choices that go beyond the content of the passage, or that deal with content only discussed in one paragraph of the passage.
3. Make brief notes – a couple of words- regarding the Main Idea on the text on your scrap paper while you read.
Primary Purpose: What is the author trying to do? What is his intention? If he is evaluating a theory, then the answer could be something like "Discuss an interpretation". Note that the correct answer would deal with "an interpretation", because the author is only dealing with one theory. If the Primary Purpose is to criticize 2 new books, then his intention or his primary purpose might be to "Critique new studies". Again, as in Main Idea questions, re-read the first line of every passage, and the last line of the first and last paragraphs. This should give you the general structure or outline of the argument, with which you can answer the Primary Purpose question.
Note: A good main idea or primary purpose does not go beyond the scope of the passage, nor does it limit itself to discussing only one part of the passage.
What is the primary purpose of this passage?
discuss the importance of the television program Star Trek for the international space program
discuss important theoretical work concerned with faster-than-light space travel.
explore a dispute among theoretical physicists regarding the uses of space flight
describe the possible uses of space-warping material
explain how a space-warping bubble would work in the real world
This is a Primary Purpose question, so we have to determine what the author is trying to do or say in this passage. So, let's read the first and last lines of the passage in order to get an idea of the primary purpose. The first line says "Great news for Star Trek fans: warp drives that can propel starships around the Galaxy faster than the speed of light may be possible after all--with a little help from Dr Who." The last line is a quote by a physicist that says "Of course, there are still some basic questions--like how does one go about constructing this Tardis space-time--but it puts the concept of space warps back on the agenda." From both these sentences, we get the idea of space travel, faster than light travel and space warps – maybe this is a discussion of faster than light space travel. Does that match what you have already read? Yes, basically this is a discussion of the theoretical state of play in the area of faster-than-light space travel. Do any of the 5 answer choices match that? Yes – B, even if the wording is somewhat different from how we are wording it, the idea is almost exactly the same. B is the answer.
Another way of getting to the answer is through elimination of obviously incorrect answer choices. We can eliminate A because the author mentions the popular science fiction program Star Trek merely to introduce the idea of faster-than-light travel, and nothing more. C is a stronger possibility because the second paragraph of the passage does discuss some disagreement among physicists about the possibility of creating a warp-drive, but in the same paragraph the theoretical dilemma seems resolved. Moreover, since the author only discusses this in one paragraph, it cannot be the primary purpose of the entire passage. We can eliminate D because the author does not go into detail discussing the uses of space-warping material. And we can discard E because the author does not really go into how the space-warping bubble would work in the real world.
by Stephen Bolton, 20th August, 1999
Title questions are very similar to Main Idea questions, though are less common. Though some of the example paassage we use in this tutorial and in the Practice Section are from the New Scientist, and therefore have titles, the passages in the real GMAT will not have titles. The question might look like this:
"Which of the following titles best summarizes the passage as a whole?"
Treat this as a Main Idea question. A good title sums up the central idea of a passage. Therefore, in order to answer this type of question:
1. Look in the first and last paragraphs for the main idea. Any conclusion words like therefore, thus, so, hence, etc. that you see are most likely introducing the Main Idea/Title. The correct answer will say the same thing as it says in the text, but using different words.
2. Re-read the first line of every passage, and the last line of the first and last paragraphs. This should give you the general structure or outline of the argument, with which you can answer the Title question.
3. Make brief notes – a couple of words- regarding the Title on the text on your scrap paper while you read.
4. After determining the general structure or content of the argument, eliminate answer choices that are too broad or too specific, i.e. answer choices that go beyond the content of the passage, or that deal with content only discussed in one paragraph of the passage.
What would be an appropriate title for this passage?
Constructing The Tardis
How To Make Space-Warping Material
Bubbles In Space-Time
Faster-Than-Light Travel: A Possibility?
Debate On The Uses of Space Travel
This passage actually already has a title, "Warp Factor One". But we have to look for another title possibility, one that would be most like the Main Idea of the passage. We look at the first and last paragraphs, and since the Main Idea is that researchers now feel that faster-than-light travel maybe more than mere fantasy, we can find the correct answer choice. Does any answer choice corrspond to this idea? Yes- answer D, which is the correct answer.
We can also find the correct answer through elimination. There is nowhere in the passage where it discusses building Dr. Who's Tardis (pity!), so we can eliminate A. Nor does it tell us how to make space-warping material. Eliminate B. While bubbles in space-time are discussed at some length in one of the paragraphs, we cannot say this is the main concern of the passage, and thus should eliminate C. And nowhere are the uses of space travel discussed, so discard E.
Specific Detail or Target questions are probably the most common types of questions, and the easiest to answer. The question might look like this:
"According to the passage,...."
"The passage states that ...."
The Specific Detail or Target that we are looking for could be a Line Number, or a Name or Date. Go to the Line Number or Name or Date, and then read several lines above and below it. Find the answer choice that basically says the same thing as in the passage, though usually with different words or word order.
According to the passage, Pfenning and Ford
demonstrated conclusively the impossibility of faster-than-light travel
explored the possibility of bubbles that warp space
supported the work of Alcubierre
work at of the Institute for Theoretical Physics at the Catholic University of Leuven
suggested that a warp drive was not physically possible
This is a Specific Detail/Target question, and therefore we look for the Name, Line Number, or Date that will help us. In this case, the detail consists of the names Pfenning and Ford. We scan the text, starting from the top of the passage, looking for the names Pfenning and Ford. We find them in only place, at the beginning of the second paragraph. We read a couple of lines above the names, and keep reading until a few lines after the names. It says "But in 1997 Michael Pfenning and Larry Ford at Tufts University in Medford, Massachusetts, apparently killed this ingenious idea by showing that it needed far more than the entire energy content of the Universe to work (This Week, 26 July 1997, p 6)". The line after that says the research of another physicist then resurrected the possibility of FTL travel, negating the implications of the research of Pfenning and Ford. Now we can answer the question. Do any of the answer choices match the information given around the target area? Yes- E.
Let's also eliminate. If we re-read what the passage says about Pfenning and Ford, we can eliminate B, C, and D. None of them are supported by the information in the passage, so let's eliminate all of them without wasting too much time and with a minimum of fuss. A is tougher to eliminate. From the sentence that mention Pfenning and Ford, it seems their work does rule out the possibility of a space-warp drive. But if we read the next line, it says another researcher said it was indeed possible. So the Pfenning and Ford could not have "conclusively" demonstrated the impossibity of the FTL drive.
This is probably the most difficult type of Reading Comprehension problem. The question might look like this:
"It can be inferred that the author makes which of the following assumptions?"
"Which is an assumption underlying the last sentence of the passage?"
"Which of the following, if true, would most strengthen the hypothesis mentioned in lines 17-19?"
"With which of the following statements regarding chaos theory would the author be most likely to agree?"
1. First, treat this type of problem as a Specific Target question. Look for a target in the question, find it in the text, and then look above and below it. Often you do not have to infer very much, the answer remains within the text.
2. If the answer must be inferred and is not stated explicitly within the text, then choose the answer choice that can be inferred or assumed from the information given. Again, you should not have to infer very much – only one or two logical steps removed from the information in the passage.
3. Make sure that the answer choice you decide on does not violate or contradict the Main Idea of the passage - if it does, the answer choice is probably wrong.
It can be inferred that a house with the propeties of the bubble mentioned in the passage
would be larger on the inside than on the outside
could move faster than the speed of light
might be very energy efficient
could move through time
would eventually fold in on itself and be destroyed
First, let's try to deal with this question as a Specific Target problem. Is there a target in the question? Yes – the bubble. The bubble is first mentioned at the end of the second paragraph, and then discussed at length throughout the third paragraph. Remember, we have to look above and below that target area (as well as read the target area again), so quickly go through the second, third, and first part of the fourth paragraph.
When you are finished, look at the answer choices. Can any of them be inferred from the information given in the target area? Well, we could eliminate C, D, and E for simply not being supported by the information given in the passage. B – maybe, but a house moving through time seems pretty silly. But in the fourth paragraph the author talks about the Tardis, "which looked like a police box but had a spacious interior". Big on the inside, small on the outside. Is that like our house? Yes- answer A. As well, we can choose A because it does not go against or contradict the Main Idea in this case, which if it had, would have made it necessary to eliminate. So choose A.
The question might look like this:
"The author's attitude towards Morgan's theory could best be described as one of ..."
Look for descriptive words, adjectives or adverbs, that could tell you the author's attitude. For example, the words unfortunately or flaw suggest a negative connotation, while strength or valuable emphasize the positive. Make brief notes – a couple of words- regarding the Tone of the text on your scrap paper while you read. Additionally, keep in mind that the author's attitude toward a theory, book, or ethnic group will almost always be respectful, even when somewhat critical.
The author's attitude towards Miguel Alcibierre's theory could best be described as one of
Since this is a Tone/Attitude question, we must look in the passage for descriptive words that tell us what the author thinks of Alcibierre and his theory. In the second paragraph the author call's Alcibierre's theory "this ingenious idea". This is positive, and the only positive answer choice is A. A is the correct answer. As well, if we could not find the tone so easily, we could also eliminate C and D at the very least, for being too extreme.
The question might look like this:
"Which of the following best describes the organization of the passage?"
"Which of the following best describes the organization of the first paragraph of the passage?"
"One function of the third paragraph is to...."
Re-read the first line of every passage, and the last line of the first and last paragraphs. This should give you the general structure or outline of the argument, with which you can answer the question. Remember to make brief notes about the structure of the text on your scrap paper. If you are looking for the organization of one paragraph, read the first and second sentence of the paragraph. That will give you a rough idea of what is the structure or organization of the paragraph.
Which of the following best describes the organization of the second paragraph of the passage?
Two investigations that support Alcubierre's theory are introduced
Possible objections to the uses of the warp drive are present, and then refuted
An objection to the practicality of the theory is raised, and then another work is cited to shore up the applicability of the original theory
A work of theoretical physics that supports Alcubierre's theory is raised, and then another that refutes it is presented
Alcubierre's theory is analyzed by a panel of several eminent physicists
Read the first sentence of the paragraph: "But in 1997 Michael Pfenning and Larry Ford at Tufts University in Medford, Massachusetts, apparently killed this ingenious idea by showing that it needed far more than the entire energy content of the Universe to work (This Week, 26 July 1997, p 6)". Then read the second sentence: "Now Chris Van Den Broeck of the Institute for Theoretical Physics at the Catholic University of Leuven, Belgium, has resurrected Alcubierre's proposal". So if we out those two sentences together, and in different words, first the usefulness of Alcubierre's theory is questioned by two researchers, then the theory is validated by yet another researcher. Which of the answer choices is closest to this? C. None of the other answer choices follow the organizational pattern of the paragraph – they reverse it, or are completely dissimilar. C is the only possible answer.
1. Read the whole text of the passage once.
2. Make brief notes about the text on your scrap paper.
3. Remember that the tone or attitude of the passage is usually respectful and moderate, never going to extremes of praise nor criticism.
4. Look out for structural words that tell you the important ideas or transitions in a passage.5. Go back to the text of the passage for the answers to specific questions.
From today onwards, I would be posting a few points on the practical usage of English. I would update it every now and then. My request to you is not to start any discussions below this thread. Please go through it on a daily basis and learn as much as you can. You can discuss the points in a separate thread.If you would like to contribute something, you can mail the material to me and I will add it with your name. My email id is email@example.com
beside and besides
Beside is a preposition meaning 'at the side of', 'by' or 'next to'
Why is the cat sitting beside the chair?
Besides is used when we add new information to what is already known.
Besides aerobics, I have to do crunches and push ups.
Besides can also be used as a discourse marker meaning 'also', 'in any case',and 'as well'. It is often used to add a stronger, more conclusive argument to what has gone before. In this case, besides usually goes at the beginning of the clause.
It's too late to go out now. Besides, it's starting to rain.
I don't like this dress; besides,it's too expensive.
besides, except and apart from
Besides usually adds; it is like saying with, or in addition to or plus (+).
Besides cornflakes, I have fruits for my breakfast.
Except subtracts; it's like saying without, or minus (-).
I like all fruits except apples.
Apart from can be used in both senses.
Apart from cornflakes, I have fruits for breakfast. (= besides cornflakes)
I like all fruits apart from apples.(=except apples)
After no, nobody, nothing and similar negative words, the three expressions (besides, except, apart from) can all have the same meaning.
He has nothing except/besides/apart from his house. (= He only has his house.)