Here are a couple more formulas you should know for the ACT and SAT...
The sum of the internal angles for a polygon with n sides (an "n-gon") is (n-2) * 180 degrees. For example, the sum of the internal angles in a pentagon is (5-2) * 180 degrees = 540 degrees.The sum of the external angles, taken one per vertex (corner) for any polygon, no matter how many sides it has, is 360 degrees. For two external angles per vertex, multiply 360 by 2 to get 720; for 3, multiply 360 by 3 to get 1080 degrees. Things to Know for the Math Section of the Upcoming SAT Here’s a quick list of formulas and other math facts you should know for tomorrow’s SAT. You will have to memorize them; they will not be provided in the “Math Facts” box at the beginning of the math sections. The “difference of squares”: (a+b)(a-b) = a2 – b2 Other binomials: (a+b) = a2 +2ab + b2 (a-b) = a2 - 2ab + b2 The Relationship of the Diagonal of a Square to its Area.A = d2/2. This comes from the fact that a square’s diagonal splits it into two 45-45-90 triangles, which have the side ratio of 1-1-√2, so the side length, s, is equal to d/√2, and the area, which is s2, is then equal to (d/√2)2, which is d2/2. Know that a square’s diagonals bisect, are congruent (equal), and are perpendicular, to each other. :Important Properties of Parallelograms- Angles that are diagonal to each other are equal.
- Consecutive angles (ones that are next to each other) are supplementary (they add up to 180 degrees).
- If one angle is a 90-degree (right) angle, then they all are.
- The diagonals bisect each other.
- Each diagonal bisects the parallelogram into two congruent triangles.
Properties of transversals (two pairs parallel lines that cross each other). Notice that parallelograms are formed by transversals (two pairs of parallel lines). Notice that angle 1 = angle 8 = angle 2 =angle 5 and angle 4=3=7=6. If there were a line parallel to line t, the corresponding angles formed with line l and m would also be equal (congruent) to the angles line t formed with lines l and m (that is, all the acute angles are congruent, and all the obtuse angles are congruent). See the first diagram in this section. Obviously, if they were all right angles, all the angles would be congruent, since they’d all be 90 degrees. Area of a Trapezoid:How to Find the Area of an Equilateral Triangle if All You Know is the Side Length:Here, if you know the side length is s, then you know you can split the triangle into two 30-60-90 triangles with side lengths s/2, s, and s√3. The s√3. side is the altitude of the triangle that splits it in two. So the area of one 30-60-90 triangle is 1/2 * s√3. * s/2, and the area of the whole equilateral triangle is (s*s√3)/4, which is (s2√3)/4. You can memorize this formula, or just know how to derive it by using an altitude to split the triangle in two.
I hope these formulas and facts help. Please read my other blog entries for other tips. Good luck on the SAT For those of you taking the August SAT, here’s some advice on how to approach the SAT essay.
First, analyze the piece of writing. It’s almost always going to be a persuasive essay – e.g., a newspaper editorial, magazine article, or similar short piece intended to convince you of something. While you are reading the piece, you should ask yourself, and answer, the following questions: - Who is the author of this piece? What does the introduction, or the piece itself, tell you about the author? For example, is the author an environmentalist or an oil company executive? How might that affect the author’s viewpoint? An oil company executive might be likely to overstate the potential gains from drilling in wildlife refuges, while minimizing the environmental damage from it (e.g. “Caribou love to rub up against the Alaska Pipeline for warmth”). An environmentalist would likely do the exact opposite (“Oil is an insanely pollutive, short-sighted solution to our energy needs, oil spills kill wildlife, and even if there were NEVER any spills, pipeline development destroys much-needed habitats and forests. Trees create oxygen. Oil wells don’t.”) Note – it is unlikely you will lose points for getting the author’s gender wrong if the name doesn’t obviously tell you the author’s gender. This isn’t a test of your knowledge of the writers’ personal lives. However, you should get such details correct if you quote another writer in your analysis of the piece.
- What is the author discussing in the piece? It shouldn’t be that hard for you to figure that out. What is the main subject – what is the author pointing out to you? Is it about war, energy, helping people, exercise, reading, or some other subject?
- What does the author want you to think about the subject of the piece? What does the author want you to think about, oh say, war, energy, helping people, exercise, reading, or some other subject? The author might want to explain why exercise is good, or why helping other people is only good to a certain extent (e.g., the author’s point may be that “helping” people too much can actually hurt them by making them lazy and dependent).
- What, if any, moral statements are expressed or implied by the author’s argument? For example, is the author implying that saving lives is a good thing (not a particularly hard idea to “sell” to most people), that exercise is good for you, that we should help those who are less fortunate than ourselves, or that people should work to earn a living? What evidence does the author use to support his or her argument? Statistics? References to well-known stories, history, current events? These items are called “ethos.” Identify all of them in your essay. Remember, the essay graders don’t know you, and will assume you know
*nothing*about the subject at hand, and give you no credit, until you prove you know something. That means you should state the obvious points as well as the non-obvious ones. Do these assumptions and supporting facts strengthen or weaken the author’s argument? Do you find them convincing? Why or why not? If you do not think these aspects are as strong as they could be, identify what the author could have done to strengthen them (assuming you can think of something). - What kind of emotional, sensory, descriptive language is the author using? This aspect of a piece of writing is called “pathos.” Identify the language and how it is used, what it is intended to make the reader think or feel, and if the use of such language is effective. Again, the SAT graders don’t know you and have to assume you’re a complete idiot until you show them otherwise. Again, don’t be afraid to state the obvious. It’s better to make a “no-brainer” point and have the grader find the essay (and you) a bit pedantic than not be credited for your knowledge of material because you assumed the grader knew that
*you*found it obvious. - What is the underlying logic of the essay? That is, how does the author use the moral and factual assumptions or statements to reach the conclusion of the essay? Or, if you prefer, how does the author make his or her point? Does the author go from the general to the specific, stating a rule, then showing how the particular situation he or she is discussing fits the rule, which means there can only be one moral and logical conclusion? That kind of logic is “deductive logic,” or a “syllogism.” (You know- syllogisms are statements like “All Smurfs are blue. Brainy is a Smurf. Therefore, Brainy is blue.”) Does the author point out the conclusion that was, or should have been, reached in one instance to generalize that the same conclusion should be reached in a wide range of situations similar to the first case he or she describes? That is called “inductive logic.” Is the logic convincing, or does the author try to skip steps or use logical fallacies such as appeals to emotion, “straw man” arguments, or the like? [See my previous entries and videos on logical fallacies for more examples, or simply Google “logical fallacies” for MANY examples].
Second, start writing. Leave several lines (probably 5 or 6) blank to write the introduction AFTER you’ve written the body paragraphs. Start your actual WRITING by addressing the subject of the piece, and what the author wants you to think of it – the point of which the author is trying to convince you. Then discuss the ethos (facts, assumptions, morality) in one paragraph, the pathos (emotional language) in another, and the logos (“logical” argument structure) in another. You can discuss the three elements in any order that makes sense to you. Alternately, you can simply analyze the ethos, pathos, and logos in each paragraph of the essay, starting with the first and ending with the last. While this takes longer and is often more boring to read, it does have the advantage of making it much less likely you’ll fail to discuss any aspect of the essay, since you’re literally analyzing it paragraph by paragraph, if not line by line. That’s the way I imagine a computer program would analyze an essay – using the “You can’t miss a spot mowing the lawn if you always mow in overlapping strips” method. Finally, if you have time and can think of some issues the author failed to address, such as obvious counterarguments, do mention them, and how the author could have made his or her argument stronger by addressing them. However, please remember it is easier to praise a piece of writing than to criticize it, since no one will make you “prove” why a generally well-written piece makes a good argument, but any grader will want to know why you think a carefully selected piece of writing the SAT used as a prompt is flawed. The grader will expect such criticism to be supported by convincing and powerful evidence and argument. If you don’t understand my point, I’ll provide an example. If I said “Hmm, those Williams sisters are pretty good tennis players, right?” most people would agree and move on to some other conversational topic. But imagine if I said “Serena Williams isn’t all that great of a tennis player.” Most people would want to know why I believe something that just doesn’t seem to be true at all. Someone would respond with something like “Um, John, she’s won many championships; she’s regarded as one of the best women tennis players in the world – what makes you think she’s not ‘all that great?’” They’d probably wait for my response while thinking “This, I’ve gotta hear…” I’d have to come up with something pretty amazing in response (which, in this case, I couldn’t do, and I’d end up looking stupid). So don’t criticize the piece unless you have a good idea of what exactly is wrong with it (for example, it relies on appeals to emotion or personal attacks, and cites virtually no supporting evidence). It’s easier to say nice things if you’re not clear on what you don’t find convincing. If you simply don’t like/don’t agree with the author’s point, but he or she has addressed your objections, you’ll simply have to discuss why you feel he or she inadequately addressed your objections. If you can’t do that, leave it alone. Also, if you AGREE with the author, don’t merely restate the author’s argument; make sure you describe how the author constructs the argument, as detailed above. I’ve seen students show they agree with, and understand, the author’s argument in their essays (e.g. why school uniforms in public school are a bad idea) by summarizing and restating it, but completely fail to address how the author states and supports the point. That got them high reading and writing scores, but very low analysis scores. Don’t be like those students. Once you’ve done that, simply write your introduction, in which you write something like "In the news editorial Why Is It So Cold In Here?, Festus Freezer makes a convincing case as to why home heating costs for the poor should be subsidized by state, local, and federal government using citations to surveys, historical and literary references, and appeals to morality (often called “ethos”), as well as powerful descriptive language and appeals to emotion (also known as “pathos”) and brings them all together with deductive logic (also known as “logos”). He dismisses arguments that subsidized heating would simply add an unnecessary tax burden to the general public and encourage the waste of heating fuel as fearmongering unsupported by any study done on the matter or any analogous study on welfare or health benefits."Finally, write the conclusion, where you restate the points you made in the introduction, and briefly list examples of, the methods used by the author. "Freezer’s argument in Why Is It So Cold In Here?” is compelling because, as I have shown, he combines facts, such as the results of several studies by the Department of Housing and Urban Development, emotional language such as “Why make your warm, loving Grandma suffer in the cold?,” and logic (mostly the argument that people should others, especially when it is not difficult to do so) to make an superlatively good argument for heating subsidies for the indigent." Then proofread your essay, make necessary changes, and relax. You’re done. Good job! Good luck on the SAT! |
## Author: John Linneball Who did you think? ;-)I'm the proprietor and only tutor for this business; that's why I named it after me. ## Archives
March 2021
## Categories |

Proudly powered by Weebly