We continued through the summer, and she worked tirelessly. For a day or two, just a couple weeks before the August test, I thought she might crack. “You don’t have to do this.” I explained. “You already have an amazing score. But if you’re going to the SAT Olympics, I’m going to coach you at that level.”
She pressed on, took the test — and got a 1,530 composite, with a 770 in English and a 760 in math. With her grades, extra-curricular activities, and a tremendous common app essay, she was accepted to her first-choice school. I bet you’ve heard of it!
Chapter 2
Review of Pre-Algebra
IN THIS CHAPTER
This chapter provides a review of pre-algebra topics you’ve probably seen before, but maybe half-remember in a fuzzy sort of way. Although some of these concepts may have given you trouble in 7th or 8th grade, you may be surprised how easy some of this stuff seems now — especially if your current math class is Algebra II or Pre-Calculus!
To begin, I discuss five key sets of numbers: natural numbers, integers, rational numbers, real numbers, and complex numbers. Then, you get a review of four ways to represent rational numbers: as fractions, ratios, decimals, and percentages.
After that, I give you a brief review of absolute value, followed by a more in-depth look at radicals (square roots). Then, I provide a clarification of the algebra vocabulary you may recognize but still be unclear about.
I finish up with a look at a short but important list of calculator moves you’ll need to know for the Calculator section of the math SAT.
Sets of Numbers
The SAT Math Test focuses on numbers that fall generally into five sets: natural numbers, integers, rational numbers, real numbers, and complex numbers. Understanding how these sets of numbers differ can be important when answering an SAT question that asks for a solution within a specific set of numbers.
In this section, I give you a brief overview of how these five sets of numbers fit together.
Natural numbers
The first set of numbers you learn as a child are the natural numbers, or counting numbers, which are the positive whole numbers starting at 1 and continuing without end:
When you add or multiply any pair of natural numbers, the result is another natural number. Another way to say this is that the set of natural numbers is closed under both addition and multiplication. However, the natural numbers are not closed under subtraction or division, because when you subtract or divide a pair of natural numbers, the result isn’t always a natural number. For example:
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Integers
The next set of numbers are the integers, which include the natural numbers, 0, and the negative whole numbers:
The set of integers is closed under addition, subtraction, and multiplication, because when you apply any of these operations to any pair of integers, the result is an integer. However, the integers aren’t closed under division, because when you divide a pair of integers, the result isn’t always an integer. For example:
Rational numbers
The rational numbers are the set of all numbers that can be expressed as fractions with integers in both the numerator and denominator. For example:
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As you can see, the set of rational numbers includes all the integers, because every integer can be expressed as the numerator of a fraction with 1 in the denominator.
The set of rational numbers is closed under addition, subtraction, multiplication, and division.
Points on the number line that cannot be expressed as fractions — such as
Real numbers
The real numbers are the combined set of both rational and irrational numbers. This set includes every point on the number line.
Like the rational numbers, the set of real numbers is closed under the basic four operations. However, the set of real numbers isn’t closed under the operation of taking a square root, because the square root of a negative number isn’t a real number. For example:
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