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SAT Math: Three Angles of a Triangle

May 18th, 2010 by Marty Rafson

Virtually every student knows that the sum of the measures of the three angles of a triangle is 180 (and for those who don’t know this, the kind folks at the Educational Testing Service have listed this fact at the top of every SAT math section).

Consider this question: What is the average measure of one angle of a triangle? It’s a simple question. To find the average of any three things, we find the sum and then divide by 3. Since the sum of the three angles of a triangle is always 180, when we divide by 3 we find that the average measure of an angle of a triangle is always 60. This fairly simple fact has nice implications for SAT math problems.

Only an equilateral triangle has 3 angles that each measure exactly 60. If we then decrease one of the angles by one degree to 59, to balance this out one of the other angles must increase to 61 so that the average remains 60 (and the sum remains 180). Now we have three unequal angles. Let’s examine them more closely. What can be said about the smallest angle of a triangle? Setting aside the equilateral triangle situation, the smallest angle of a triangle must measure less than 60. Are there any other conclusions that we can draw about the other two angles of the triangle? The largest angle of the triangle must measure greater than 60. Is there anything we can state for sure regarding the “middle-sized” angle of a triangle? It may be more or less than 60, we can’t tell for sure. However, if you give it some further thought, could this angle be as large as 90 degrees? If the “middle-sized” angle of a triangle is 90 and then the largest angle is more than that, we would have more than 180 degrees. Therefore, the “middle-sized” angle must measure less than 90.

All of these conclusions derive directly from the simple fact that the sum of the measures of the angles of a triangle is 180. Make no mistake about it – the SAT will test you about these more in-depth concepts. Taking an SAT Prep Course will hone your math skills.

Marty Rafson wrote the ESC math curriculum and has been an SAT math teacher, tutor, and curriculum developer for 30 years. He has been a high school math teacher for 36 years and a math department chairman for 25 years. He was also an adjunct professor at City College of New York School of Education.

Tags: SAT Math, SAT Prep, SAT Preparation
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SAT Math: Factoring the Difference of Two Perfect Squares

May 10th, 2010 by Marty Rafson

Every Algebra 1 course includes a unit on factoring. When teaching students, we usually break down factoring into three types: Factoring the greatest common factor; factoring the difference of two perfect squares; factoring trinomials. The SAT will not simply come out and ask a classroom-type question requiring the student to factor a given expression. An SAT problem will require more cleverness as the need to factor is often hidden in the problem. In fact, it is rare for an SAT math question to even give a hint that the student needs to factor an expression. Of the three types of factoring, experience shows us that factoring the difference of two perfect squares has shown up the most on past SAT’s.

Problem: “If x + y = 10 and x – y = 2, find the value of x² – y².” There are two interesting mathematical approaches possible here. As we read the given information, it appears that we have to solve a system of two equations to find the values of x and y and then plug them into the given expression x² – y². However, often on the SAT, it is critical to focus on the right part of the problem. x² – y² at the end of the problem should leap off the page and catch the attention of the sharp math student. This expression screams to be factored as (x + y)(x – y). Any student who notices this will then immediately notice that we have been given the values of each of these. The first is equal to 10 and the second is equal to 2 and all we have to do is multiply to get the answer of 20.

Would we have reached the same result if we had solved for x and y? Absolutely! Solving the system of two equations would have yielded x = 6 and y = 4. Then, 6² – 4², or 36 – 16 would have given us the correct answer of 20. However, how long would that have taken? Remember, on the SAT the clock is ticking. Recognizing the factoring of the difference of two perfect squares yields an almost instantaneous answer. Further, any student seeing this should recognize that he/she has just outsmarted the person who wrote the question. That just feels good and keeps your spirits high as the test is going on. Keep up your SAT prep for a great score!

Tags: SAT Math, SAT Prep, SAT Preparation
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Choosing Your Senior (or Junior) Year Courses

May 5th, 2010 by Joe Connell

As I wrote in the blog about “College Admission Requirements,” one of the key college admission factors that sophomores and juniors can still manage in addition to their SAT scores is the course schedule they create for their next academic year. Thispost provides some information to keep in mind, when choosing next year’s classes.

Take as many honors, AP, IB and college-credit courses (often offered through local colleges) as reasonable for you to be successful. Many colleges take weighted GPAs for determining admission, so if your high school weights GPAs, a “B+” in AP US History is often going to strengthen your college application more than an “A” in regular high school history. In addition, many competitive colleges provide their own “weights” to courses considered to be college-level. Therefore, taking more advanced courses can help a student twice in the admissions process – i.e., both in the high school GPA and the college admissions review of your high school transcript.

Here’s a breakdown of recommendations by subject area:

Complete a minimum of four years of English.
Complete four years of mathematics. (These courses should become more challenging junior and senior year. If possible, move on to trigonometry, pre-calculus, calculus, statistics, etc.)
Complete at least three years of social science (history, government, social studies, psychology, economics, etc.).
Take three years of laboratory science. (Preferred courses include: Chemistry, Physics, Biology, Anatomy and Physiology, AP courses in any of the aforementioned.)
Complete at least three years of a foreign language. (Four or five years will really make you stand out.)

If you are not sure which courses are best for you, talk with family, friends and high school staff, especially your guidance counselor, about your college plans and courses that they recommend.

Joe Connell has been helping high school students transition to college for the last nine years through positions in admissions, new student orientation and retention. Currently, Joe is the Director of Academic Services & Testing at Dutchess Community College (NY); he has previous work experience at William Paterson University (NJ), Marist College (NY) and Miami University (OH). Joe has presented on issues related to college students' transition and success at both regional and national levels and has taught both SAT preparation and college courses for the past eight years.

Tags: College Admissions, SAT Prep, SAT Preparation, SAT Scores
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SAT Algebra

May 3rd, 2010 by Marty Rafson

Back in Algebra 1, all students were assigned problems such as (x + y)². At that time most students were advised to write the (x + y) twice, and then multiply. Many teachers, when multiplying polynomials, employ the FOIL pneumonic device. “First”, “Outers”, “Inners”, “Last”. This would give us x² + xy + xy + y². Simplification yields x² + 2xy + y². The 2xy is referred to as the “middle term”. Some teachers, perhaps when the students reach the Algebra 2 level, ask the students to memorize this form. Every time we square a binomial we are going to get two identical middle terms from the “outers” and the “inners”. Of course, the less adept math students will look at (x + y)²and merely get x² + y². The writers of the SAT, the Educational Testing Service, know how to test students to be sure that they know about the existence of the middle term.

Problem: “If a² +b² = 13 and ab = 7, find the value of (a + b)².” If you memorized how to square a binomial, you will instantly write down a² + 2ab + b² (of course, if you are not confident with this, you can write down a + b twice and “FOIL” it). Note how the given information perfectly fits this form. We are given that the first term plus the last term, a² +b² has a value of 13. Now, examine the middle term. If ab = 7 then 2ab must have a value of 14. Thus, the value of the given expression is 13 + 14, or 27.

Knowing how to multiply polynomials is a must in any algebra course and is important to practice for the SAT. Specifically, knowing how to square (x + y) quickly can be a great asset on the SAT.

Tags: SAT Math, SAT Prep, SAT Preparation
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SAT Test Day is Almost Here. Now What?!

April 27th, 2010 by Kate Hedman

The May SAT is nearly upon us, and some of my students recently asked me why they’ve heard that the SAT takes about five hours when the College Board states that it’s only three hours and fifty minutes long. That question, as well as the fact that the test is this weekend, made me think that I should share some last minute tips and information with you.

First of all, the SAT really is a three hour and fifty minute test. The College Board doesn’t sneakily add time to it or anything. But they do give you three five-minute breaks. So there’s an additional fifteen minutes. They also ask that you arrive about an hour early, or by 7:45am. You’ll get checked in and get an assigned seat, and the test center doors close at 8:15. No students are admitted after that, so BE ON TIME! Basically, it’s the test plus all that extra stuff that adds up to about five hours of being at the test site.

During the breaks, you will be allowed to eat and drink, so it’s a good idea to bring snacks and drinks with you. One snack that helps you improve your test scores is chocolate, which is good news for anyone out there who loves the stuff. Supposedly, it’s the flavinoids that make the difference, so if you don’t like chocolate, have some tea, which may have a similar effect.

In the last couple of days before the SAT, do a little bit of sat prep studying, and double check the College Board Checklist to make sure you have everything you’ll need. The night before the test, put out your clothes and supplies for test day and get some good sleep. You want to be organized and well rested so that you can focus and do your best. On test day, eat a good breakfast, get to your test site early, and remember to focus on doing your personal best. Good luck!

Kate Hedman, MSEd, has been helping students succeed on the SAT for seven years. She has been a verbal teacher with ESC for six years, and taught high school English for three years. She loves reading about new advances in brain research that she can use in the classroom to help her students learn how to achieve higher scores on the SAT.

Tags: SAT Prep, SAT Test, SAT Test Day
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SAT Math: Shortest Distance Between Two Points

April 26th, 2010 by Marty Rafson

Euclid is credited as the founder of most of our modern geometry. Though he lived approximately 300 years B.C., his foundation of geometry is still the basis of the geometry we teach in high school and, therefore, the basis for SAT questions on geometry. One of the most common geometry questions on recent SAT’s goes right back to one of Euclid’s most basic ideas.

What is the shortest distance between two points? Euclid said the shortest distance between two points is a straight line. This was one of his fundamental axioms, or postulates – statements that are accepted without proof because they are so obviously true.

Consider this situation: You need to travel from point A to point B. According to Euclid, the shortest distance would be along the straight line AB. However, now let’s suppose that you had to divert first to a point C that is not along this straight path. You would first travel from A to C (AC) and then from C to B (CB). Clearly, this second trip which takes us first to point C is the longer trip. If you draw a triangle to represent this situation, we can rename the result: AC plus CB must be longer than directly from A to B, namely AB. Thus, based on Euclid’s axiom, we have proven a theorem: The sum of two sides of a triangle must be greater than the third side. Someone at the Educational Testing Service has decided that this is a critical concept and a question on it has appeared on most recent SAT’s.

Problem: “Two sides of a triangle are 5 and 9. If the length of the third side is an integer, find one possible length of the third side.”

Let’s try 3. This doesn’t work since 5+3=8 which is not greater than 9. Perhaps we should try 4. Now, 5+4=9 but is not greater than 9 so this is not good. If we try 5, the sum of any two sides is greater than the third side. Note that 6,7,8,9,10,11,12,13 also work. When we reach 14, 5+9=14 but is not greater so that doesn’t work.

Problem: “Two sides of an isosceles triangle are 2 and 5. Find every possible perimeter of the triangle.” Isosceles triangles have two equal sides. So, the triangle could have sides of 5, 5 and 2 giving us a perimeter of 12. However, many students would then suggest that there’s another isosceles triangle with sides of 2, 2 and 5. Sadly, they would be wrong. Since the sum of any two sides must be greater than the third side and 2+2 is not greater than 5, this second triangle doesn’t even exist. The only possible perimeter is 12.

The sum of any two sides of a triangle must be greater than the third side. Make sure you know this when you prep for the SAT.

Tags: SAT Math, SAT Prep, SAT Preparation, Test Prep
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Why Should I Prepare for the SAT Test?

April 22nd, 2010 by Joe Connell

When I teach SAT preparation, I tell my students that I am not only helping them improve their SAT score, but also increasing their chances of college admission and scholarship. The SAT is a test designed to provide a national benchmark on college readiness (a factor that the admissions office reviews for each school). As such, students should view the SAT as a tool to help them achieve their educational goals.

When approached from this perspective, students create the mindset that the SAT is a resource for them to get into that dream college. I find this outlook takes a lot of pressure off students and often motivates them to treat the SAT like part of a game. The higher the SAT score that they earn will lead to more points for the game of getting into college.

To achieve the greatest score in the SAT game, students should prepare. Students will improve their chances for success on the SAT through preparation, instruction and coaching. To provide an analogy to sports, the most successful athletes are often the hardest workers (for examples, think of Peyton Manning, Jerry Rice or Michael Jordan). Any great performer knows that talent takes you to a certain point, but effort enables you to achieve your highest potential. If you are looking for effective SAT preparation to achieve your best SAT score, Educational Services Center offers both traditional classroom SAT prep and online SAT prep. If you take the online SAT prep course, you might even meet me as the Critical Reading and Writing instructor.

In sum, a student should prepare for the SAT to increase their chances of getting into the college of their dreams!

Tags: College Admissions, SAT Prep, SAT Preparation, Test Prep
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A Multiple Choice SAT Writing Test?

April 21st, 2010 by Kate Hedman

Have you ever taken a multiple choice writing test? Seems like an oxymoron, doesn’t it? But that’s exactly what you’ll find on the SAT. As a matter of fact, the dreaded essay is only worth a third of your writing score. The remaining two thirds of your score comes from how well you answer the multiple choice questions, so it’s worth your while to put more time into working out a strategy for answering these questions than it is to spend time worrying about how well you’re going to do on the essay. There. Feel better now? Okay, then let’s get started on some question strategy.

First of all, two of the three types of questions in the Writing multiple choice section, the Improving Sentences questions and the Identifying Sentence Errors questions (which make up the bulk of the section) go from easy to hard. Therefore, the first few of each of those types will be easy (most students will get them right) then the next few will be medium (some to many students will get them right) and the last few will be straight-up difficult (few students will get them right). That means that when you’re working through these types of questions, it’s a good idea to adjust your strategy as to the difficulty levels of the questions. As the questions begin to get more challenging, you should start using process of elimination, strategically guessing, and even skipping questions.

The third type of multiple choice question in the Writing Section is the Improving Paragraphs question. These questions come at the end of the longer multiple choice section, and like the Passage-Based reading, they go in order of the passage, not from easy to hard. So you should skip around when you get to these, answering them quickly and skipping those that are taking you too long. Some students who move more slowly through the multiple choice find it helpful to tackle these first, as they tend to be a good place to get points.

A general rule of thumb for pacing on the Writing section is that in order to get a shot at answering each question, you should budget yourself about fifty seconds for each Improving Sentences question, thirty seconds for each Identifying Sentence Errors question, and a minute (including reading time) for each Improving Paragraphs question. But remember, these are just general guidelines, and are no substitute for SAT practice, which will help you figure out how to tweak your pacing in order to get your best possible score.

Tags: SAT Prep, SAT Preparation, SAT Strategies, SAT Writing, SAT Writing Section Help
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Averages – the SAT Way

April 19th, 2010 by Marty Rafson

Every student knows how to find his/her test average. If you’ve taken 5 tests, just add them up and divide by 5 to find the average, or mean. Such a straightforward problem simply won’t appear on an SAT. Instead, one of the best SAT strategies involves how to deal with the “backwards” situation.

Consider this problem: “Three numbers have an average of 50. Two of these numbers have an average of 40. What is the third number?” If three numbers have an average of 50, what must their sum have been? Of course, this sum must have been 3 times 50, or 150. You should train yourself to jot down 150 even before reading the second sentence. This is always the key to such a problem on averages. Use the same method with the second sentence. If two numbers have an average of 40, their sum must have been 2 times 40, or 80. Now, to find the missing third number, just subtract 150 minus 80, yielding 70.

This strategy can be used in a very common type of problem on averages. “On her first 5 math tests, a girl has had test scores of 88, 93, 84, 87 and 89. What must she score on the sixth test in order to pull herself up to a 90 average?” Employing this method, we ask ourselves how many total points must she have after 6 tests in order to have a 90 average. Simply multiply 6 times 90, or 540 points. Adding up her first 5 test scores we see that she had accumulated 441 points. Just subtract 540 minus 441 telling us that she better do some serious studying since she needs 99 points on this sixth test.

Approximately 30% of the SAT math questions concern arithmetic concepts – no algebra or geometry involved. One of the favorite arithmetic topics is averages and this method is often used once or twice on every SAT. I often tell SAT Prep Course students that if the first sentence of the first math problem of their SAT says “Five numbers have an average of 8,” they should immediately write 40 (5 times 8 ) on their paper and this will surely lead to an easy solution.

Tags: SAT Math, SAT Prep, SAT Preparation, Test Prep
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Help for Passage-Based Questions on the SAT Critical Reading Section

April 15th, 2010 by Kate Hedman

There is a very simple way to deal with long, difficult, and boring Critical Reading passages on the SAT: don’t read them. Surprisingly, you can answer all of the questions correctly without reading any of the long passages from beginning to end, and instead using the line references to guide you. Remember, your goal when taking the SAT is to get as many points as possible. You’re not reading for pleasure, you’re reading to find the answer to specific questions posed by the test-makers. So it makes sense to use their questions to guide your reading.

When faced with a long critical reading passage, you should follow a series of steps. First, read the blurb in italics. This will give you valuable contextual information about the passage. Second, look for line references in the questions and answer only questions that have them, skipping any that either lack line numbers or ask about the passage as a whole. Third, go back and answer any questions without line references. Let’s look at those steps in more detail, using an example from the College Board’s website.

Step One: Read the italics. In our example passage, the italics state “This passage is an excerpt from a work published in 2000 by a Chinese American writer.” We didn’t learn too much here, but we do know that this is a recently written work by a person with a mixed cultural background.

Step Two: Answer questions that have line references. In order to do this with the example passage, we are going to skip the first question, because although it has a line reference, it is asking for what “the passage as a whole suggests.” We can’t answer that yet. We can, however, answer the second question by reading lines 75-81. The lines in the line reference are clearly describing how well this Chinese American is fitting in in China, so the answer must be D. Similarly, we would use line references to answer the third question. We need to read the lines around line 53 to learn that the “revelation” is the fact that everyone around the person in question is similar to him or her. To answer the question correctly, we need to figure out which of the given lines shows that statement to be untrue, checking out each answer to see which one accomplishes that. The correct answer is E, because the “epiphany” is that he or she does not fit in.

Step Three: Go back and answer any questions about the passage as a whole. We skipped the first question because it asked what “the passage as a whole suggests” about a specific line. Now we can use what we’ve learned in the line references to answer it. Because we read lines about the person fitting in in China followed by lines that stated that the sense of belonging was false, the answer that makes the most sense is that being a “citizen of the world” is D, “an unrealistic goal.”

A lot of the passage went unread when we used this technique to answer the questions, and if we have extra time at the end, it is perfectly reasonable to go back and read some more of it to gather further evidence for our answers. However, it is important to note that we are reading for just that: to gather evidence for our answers. We need read no more of the passage than we need to answer the questions. Keep your goal in mind: to get as many points as possible. And use these steps when approaching a long passage. You will save time and increase your SAT score.

Tags: Passage-Based Reading, SAT Critical Reading, SAT Prep, SAT Preparation
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