Foci of an Ellipse

I read an article on local math whizzes who visited President Obama in the Oval Office.

They asked him "where are the foci of the Oval Office?"

It turns out, finding the foci of an ellipse is another extension of the Pythagorean theorem. As seen below.

Notice the 'a' 'b' and 'c ' here are not as they usually are in the Pythagorean theorem. "C" is a leg of the triangle here, not the hypotenuse. Otherwise, the formula they give here is just to find the leg of a triangle (here called "C" and falling on the X-axis), using the co-vertex and origin to make a right triangle.

Also notice that in this example, they use a 3-4-5 Triangle, which is a Pythagorean Triple.

If you want to read some academic history on the further wonders of Pythagoras, you can stretch your mind here.

A Pythagorean Triple

A landscaper is constructing a rectangular garden. The width of the garden measures 9 feet and the length is 12 feet. What should the length of the diagonal measure be to ensure that the garden is a rectangle?


This questions asks for the hypotenuse of a triangle with legs of 9 and 12. You could square both numbers and sum them, then take the square root.

The better, and faster way, is to recognize the 3,4,5 pattern and, in your head, quickly figure the hypotenuse to be 15.

This question came from a PSAT practice test.

A Taste of Geometry

Pythagorean Triples: Certain sets of numbers have a very special property in connection to the Pythagorean Theorem.

Not only do these numbers satisfy the Pythagorean Theorem, but any multiples of these numbers also satisfy the Pythagorean Theorem.

For example: the numbers 3, 4, and 5 satisfy the Pythagorean Theorem. If you multiply all three numbers by 2 (you will get 6, 8, and 10), these new numbers ALSO satisfy the Pythagorean theorem.

The special sets of numbers that possess this property are called Pythagorean Triples. The most common Pythagorean Triples are: (3, 4, 5) and (5, 12, 13), and (8, 15, 17).

The first two numbers are the sides of a triangle and the third number is the hypotenuse. Thus 3^2+4^2=5^2.

These are important numerical relationships to remember, as they occur frequently on the SAT.

The key is that if you see any triangle with two sides that are proportional to say a 3, 4, 5 then you can quickly predict the other side. So if you are given a triangle that has a hypotenuse measuring 20, and a side measuring 12, you can tell that it is similar to a 3,4,5 triangle and the third side must be 16. Knowing this saves the time of running the numbers through the Pythagorean theorem. And that saves you time for more difficult questions.

Prepositions

• Here is a partial list of prepositions. Remember that a preposition is neither a noun nor a verb, hence it cannot be either a subject or predicate. Try to make sentences using these prepositions, and you will find that a noun or pronoun always follows a preposition. (sometimes an article and/or adjective may lie between the preposition and the noun). Do you remember what an 'article' is? A, an, the.
• 
  
• aboard
  
• about
• above
• across
• after
• against
• along
• amid
• among
• anti
• around
• as
• at

• before
• behind
• below
• beneath
• beside
• besides
• between
• beyond
• but
• by

• concerning
• considering

• despite
• down
• during

• except
• excepting
• excluding

• following
• for
• from

• in
• inside
• into

• like

• minus

• near

• of
• off
• on
• onto
• opposite
• outside
• over

• past
• per
• plus

• regarding
• round

• save
• since

• than
• through
• to
• toward
• towards

• under
• underneath
• unlike
• until
• up
• upon

• versus
• via

• with
• within
• without
• You can find a decent explanation of how to use prepositions here: http://www.writingcentre.uottawa.ca/hypergrammar/preposit.html

13 Verb Tenses

Here is a review of the verb tenses we studied this week.

Simple Present: I live in San Jose.

Present Continuous: I am living in San Jose.

Simple Past: I lived in San Jose.

Past Continuous: I was living in San Jose.

Future: I am going to live in San Jose.

Future: I will live in San Jose.

Future Continuous: I will be living in San Jose.

Present Perfect: I have lived in San Jose.

Present Perfect Continuous: I have been living in San Jose.

Past Perfect: I had lived in San Jose.

Past Perfect Continuous: I had been living in San Jose.

Future Perfect: I will have lived in San Jose.

Future Perfect Continuous: I will have been living in San Jose.

Same idea, different period in time

More Essay Critique

Three more points from your first essays:

1. If you ask a "rhetorical question," you need to punctuate it with a question mark. One example you used: It is a good thing to do, because what is the point of keeping that joy inside you?

Rhetorical questions are used when you pose a hypothetical question to your reader, and you assume that the reader will answer in the same way you do. You assume that they agree with you, and you want to use that presumed agreement to further persuade them of the larger thesis your essays defends. Be careful not to overuse this rhetorical device; people do not always agree with you.

"rhetoric" in general is the art of communication -- particularly persuasion. How to make others believe or agree with you when writing essays or speaking in public. There is a long history of rhetorical strategies in western history, beginning with Greek and Roman orators -- "rhetorical questions" being just one example.

Here is a link on rhetoric. It's good to know about, both for writing and for understanding the foundations of Western history.

2. Do not write the way you speak. Writing is a formal exercise that requires more effort to do correctly. Speaking is done on the fly, without always following grammar rules as closely as proper writing does.

Sometimes the difference can be as simple as using a comma at a point where, when speaking, you would pause.

Two examples from a past essay: When you do good, let it go. When you do bad, talk to a friend and let it go.

Notice that "when you do good" and "when you do bad" are both adverb clauses. "you" is the subject and "do" is the verb in both adverb clauses. The subject of the independent clause is implied, given that the verb is in the imperative (command) form.

In speaking we often pause, while in writing such moments are marked by punctuation, such as a comma.

Pauses also mark transitions between grammatical structures that do not require punctuation, for instance before and after a prepositional phrase. For example: The van {with the raft on top} is mine. The natural place to pause here is after "top" which is the end of a preposition and immediately preceding the man verb of the sentence "is." The subject of the sentence is "van." Paying attention to these non-punctuated pauses can help when analyzing the grammar of a sentence.

3. Don't be repetitive. A certain amount of limited repetition is necessary. In the beginning you introduce your main ideas, in the body you develop your main ideas, and in the conclusion you briefly reiterate your main idea. The key to avoiding sounding repetitive is to thoroughly develop your abstract ideas with concrete examples and details. This way the abstract idea is repeated only in key spots, like the end of the introduction and the topic sentences of the body paragraphs, all the other sentences are using different language to develop and exemplify the abstract statements.

Also the main idea is always linked to the function of the paragraph it is in. So in the introduction the main idea is presented as an answer to the question. In the conclusion, the main idea should be linked to some broader idea that transitions the reader out of the narrow treatment of the idea that you offer, and into the broader world in which your opinion, and the question, operates.

Examples of my abstract idea, I am afraid, will have to wait until we meet again and discuss this common approach to writing introductions and conclusions.

SAT Essays

A few things to keep in mind about SAT essays:

1. Critical thinking - you need to offer a logical opinion, supported by valid reasons.
2. Develop one main idea, your thesis. Offer multiple defenses and exemplifications of that one central idea.
3. Use clear language. This doesn't mean simple language; it means one should not be confused about what you are trying to say. The grader of your essay will spend very little time reading your essay. They do not want to ponder deeply on your essay, they want to understand it on the first pass.
4. Vary your sentence structure. Use a mixture of longer complex sentences, and shorter direct sentences. Don't do this simply for the sake of variety, although that has some value in itself. Do it to help make your point clearer, and your position stronger.
5. Make sure the examples you use to develop your main idea are clearly appropriate (they are on topic, not just tangentially related to your topic).
6. Your thesis should not be obvious; it needn't be amazingly insightful, but it should not be a cliche'. It should be contentious enough that someone could have a different point of view, while still respecting the merits of yours.

I'll give you many more such posts on essay writing in the future. This is just a first run that introduces some of the vocabulary that SAT evaluators use when describing a level 6 essay.