Perfect 800: SAT Math: Advanced Strategies for Top Students

2013

Introduction: Why Another SAT Book?

The real voyage of discovery consists not in seeking new landscapes but in having new eyes.

—Marcel Proust

The vast majority of SAT-prep books are structured as one-size-fits-all products. As a result, (a) less-than-average and average students find themselves emerged into a system that does not provide them with an easy strategy to improve (i.e., identify and work on their weaknesses) and (b) advanced students would find a significant percentage of the covered topics to be too easy and thus of insignificant—if any—value to them.

This book is designed to primarily address the needs of gifted and advanced students (i.e., students expected to score above 600 on the math section of the SAT, as shown in Figure 1) by showcasing various math topics via (mostly) hard¹ problems. In addition, it offers an approach to studying that emphasizes the importance of logical thinking and the idea of reasoning as paramount to problem solving.

Figure 1. Graphic representation of SAT Percentile Ranks for 2012 college-bound seniors—total (male/female)-data published by the College Board (2012; see http://media.collegeboard.com/digitalServices/pdf/research/SAT-Mathemathics-Percentile-Ranks-2012.pdf).

This book’s methodology not only will ensure that every topic is well-suited for its audience of gifted and advanced students but also, by emphasizing critical thinking/analytic skills, will result in a much more optimal usage of your time and maximize the pace of progress in preparing for the test.

This Book’s Approach

The world we have created today … has problems which cannot be solved by thinking the way we thought when we created them.

—Albert Einstein

Scattered throughout the book are 200 problems solved as test cases. They were selected for (a) being representative for an entire category of problems that historically were given in real tests and (b) their suitability for showcasing the importance of logic and analytic skills over memorization.

Every time an example is given, students are strongly encouraged to take a few minutes not only to familiarize themselves with the problem but also to try to solve it. Only after that should you go over the solution suggested by the author. Note that almost without exception, SAT problems have unique solutions² obtained through a variety of methods. Selecting and/or suggesting the quickest or the smartest method is a very subjective endeavor. It depends on a variety of personal traits and skill sets including, but not limited to, the prevalent memory type (visual or abstract), whether you are naturally inclined to rely more on logic or sheer memorization of factual knowledge or formulas, familiarity with algebraic techniques, knowledge breadth, and so forth. As a result, it is not my intention to suggest that my way of solving a problem is either (a) the only way or (b) the best way. It is, however, my intention to use my approach as a brainstorming exercise that would eventually act as a catalyst in helping you develop your own problem-solving approach and methodology.

This book includes an online test package (see http://www.prufrock.com/perfect800) with a simulated test that, once completed by you, will generate an in-depth analysis of your results and a list of recommendations for an improved performance.

My philosophy and approach are eloquently summarized in the words of Albert Einstein: Any fool can know. The point is to understand.

1 In a typical PSAT, approximately ⅓ of problems are rated easy, ⅓ medium, and ⅓ hard. This also is valid in the SAT test, where problems are ranked on a scale from 1 to 5 with 5 being the most difficult.

2 The rare exceptions I found are problems for which two numerical solutions are equally valid.

Part 1: General Test-Taking Advice and Strategies

Chapter 1

What Is Math and Why Do I Need to Study It?

Math—the language nature uses to describe her wonders … Math is nature’s playbook, a tool for discovering—and not merely describing—nature’s laws.

—Albert Einstein

Mathematics is the discipline that deals with concepts such as quantity, structure, space, and change. It evolved, through the use of abstraction and logical reasoning, from counting, calculation, measurement, and the study of the shapes and motions of physical objects.

Math takes relevant elements of factual knowledge and, using analytic or critical thinking skills, combines them to solve problems including (believe it or not) those of the following type:

Professor: Suppose I know three people. The result of multiplying their ages is 36. Your house number equals the sum of their ages. Can you figure out how old each one of them is?

The student grabs a pencil and starts scribbling numbers on a paper pad. Two minutes later, he goes back to the professor.

Student: Are you sure you gave me enough information to answer your question?

The professor, after repeating out loud his own words, replies: Actually you’re right. I forgot to tell you that the oldest one plays the violin.

The student looks back at his worksheet and immediately, with a sigh of relief, gives the professor the correct answer.

(Find the solution in the Appendix on p. 265.)

Why you Should Never Say I Hate Math!

Nature is the realization of the simplest conceivable mathematical ideas.

—Albert Einstein

As one might infer from the previous information, math is a set of essential life skills in as much as it allows us to better understand, make sense, and take advantage of the myriad of aspects of the reality in which we live. Real life presents itself as a collection of mundane problems and our ability to solve them plays an essential role in our quest for happiness.

For example, writers, based on their work habits and history, may have to give their publishers an educated guess to help estimate the completion date of their manuscripts. French teachers on field trips to France should use basic arithmetic skills to give students a lecture in home economics with a local flavor, taking into account currency conversion and an understanding of the local tax system. Travel agents might want to be able to estimate the distance between two locations using available maps. Pharmacists, to prepare different dosages of the same medication, would need to understand and apply rules of proportionality to figure out the quantities of various required substances and ingredients that constitute the components of a prescribed drug. Police detectives need good critical thinking skills to put together disparate pieces of the case they are trying to solve and be comfortable with a thinking process that requires multiple iterations and multitasking. Physicists and computer scientists use math concepts as tools that allow them to understand the laws governing physical processes and the machine language and functionality of their computers. And, homemakers can save money by collecting coupons, becoming intelligent buyers, and optimizing their household budgets with good knowledge of and application of concepts such as retail cost, wholesale cost, percentage of discounts, local taxes, shipping and handling charges, and hidden costs.

In its importance, math—being also the science/art of how to put one’s knowledge to work in real life—transcends the borders of science and engineering into humanities and all other walks of life. So, even if your goal is to become a history teacher, you will still need math!

Chapter 2

Test Preparation Strategies and Advice

How Too Much Strategy Can Be Hazardous to Your Performance

There is only one corner of the universe you can be certain of improving and that’s your own self.

—Aldous Huxley

There is far too much printed advice on strategy in approaching test preparation. For the SAT math section, unfortunately, a large percentage of it is based on pseudo/quasi-scientific methodologies such as how to guess right, how to solve problems via trial and error, and similar tropes. The danger of emphasizing the importance of strategy is that it shifts attention from the fundamental thing that is required in math preparation—the realization that there is no substitute for the need to acquire the knowledge and skills that are essential in problem solving.

My extensive experience in test preparation has revealed three major areas on which test scores depend: factual knowledge, analytic/critical thinking skills, and concentration skills.

Factual Knowledge

There has not been, is not, and will never be a consensus on how much memorization of factual knowledge is necessary for a well-rounded education. There certainly is a decent body of knowledge that you are expected to master in preparation for the math section of the SAT tests. That begins with simple concepts (e.g., arithmetic operations, definitions of two-dimensional geometric figures, etc.) and ends with more sophisticated math concepts such as compound probabilities. Average and above-average students are not expected to have significant gaps in their factual knowledge or math background.

Analytic/Critical Thinking Skills

One may refer to these as the glue that keeps together our body of knowledge and the catalyst that gives us the ability to put it to use in solving real-life problems. This is what is missing when a student cannot solve a problem and yet has the same theoretical background as someone who shows him how to solve it. Gosh, how come I didn’t see that? is the typical reaction, noticing that finding a solution did not require the use of a magic formula or, for that matter, any factual knowledge with which the student was not familiar. Possession of these skills is paramount to achieving excellence in any profession, not only in those with mathematical or scientific/engineering orientations. These skills are the main reason why nobody should ever be justified in downplaying the role of math in our general education.

The following problem outlines the importance of analytic skills in problem solving.

Problem 1

Given the triangle shown, select the correct answer from the following:

(A) Area of triangle is less than 75

(B) Area of triangle is equal to 75

(C) Area of triangle is greater than 75

(D) Area of triangle is either greater than or equal to 75

(E) There is not sufficient data to calculate the area of the triangle.

Solution 1

Note that this is basically a quantitative-analysis-type problem³ that requires that we compare a variable, the area of the triangle, with the number 75.

After drawing the height of the triangle (h), the formula that gives us the area of the triangle is:

The steps described below lead us to a simplification of the problem whereby what we are left to compare is "h and 5 (much better/simpler task than comparing area of triangle and 75"). The following comparisons are equivalent:

The notations we added to the figure, emphasizing the fact that "h is perpendicular to the base of the triangle, should easily point out the fact that the two elements we need to compare are the sides of the newly created right triangle. Because h is a side and 5" is the hypotenuse, and because the latter is longer than any of the sides of the triangle, it follows that

h < 5

and thus the area of the triangle < 75 and the answer is A.

Note that the only factual knowledge required to solve this problem was the formula for the area of a triangle and the property of right triangles according to which, the hypotenuse is the longest of the sides.

Concentration Skills

To understand the role that concentration plays in taking the SAT test, try to answer the question: "When is the last time you sat down and worked on anything for almost 4 hours⁴? Considering that for most students, a candid answer is never," one should wonder how come there is so little interest in or emphasis on this aspect of test preparation in most preparatory books available on the market.

In my experience, this is the primary reason for underachievement by above-average students (a category that includes the not-doing-well-on-tests type).

If someone wants to run a marathon, that person needs to prepare/practice for it! No matter how good of an athlete a student is in a different sport, running a marathon requires a unique combination of skills in the areas of endurance and effort optimization.

I suggest two ways to address this issue and improve one’s concentration skills:

•   A time window equal to the duration of the test (3 hours 45 minutes) should be periodically allotted to taking a complete test (SAT Verbal and Math). That, of course, would require a distraction-free environment (e.g., no phone calls, background music, snacks, etc.).

•   Students should change the approach they use in dealing with their homework. Imagine that all homework (say, preparation for a chemistry test, some reading for geography, and a couple of exercises in French) would be treated as one monolithic task. That would require the same distraction-free environment and need to work nonstop until everything that has to do with the next day’s homework is accomplished. In this way, not only the quality of homework is expected to increase (with a positive impact on grades) but also the students will