Upper Level: Mathematics Achievement Study Guide for the ISEE

General Information

This math section of the ISEE is centered on your ability to “work out” problems, using procedures and calculations. It will be necessary to know common mathematical procedures to do the calculations. Knowing methods of checking your calculations will also help. You will not be able to use a calculator or scratch paper, but you may make notes in the test booklet.

Before test day, take some time to review the concepts and procedures below. You may want to seek additional online or paper/pencil practice with anything that gives you trouble, just to boost your memory. Keep in mind, however, that the school really needs to know the level at which you can do math, so trying to teach yourself new things just for the test will prove counter-productive to your overall school success.

The best preparation for this test would involve seeing how much you remember about the following concepts and the procedures involved in calculating within them.

Numbers and Operations

Representing Very Large and Very Small Numbers

Word problems that involve very large or very small numbers will usually represent these numbers using scientific notation. This is the standard manner of writing numbers of this kind and follows this form:

\[N × 10^a\]

Where \(N\) is a number more than \(1\) but less than \(10\) (\(1 \lt N \lt 10\)) and \(a\) is a positive or negative exponent representing how many times and in which direction the decimal point was moved to get to the number \(N\).

An item may ask for the evaluation of numbers written in this notation, for example:

Solve for the value of:

\[\frac {2.43 × 10^8}{2 × 10^{-3}}\]

Types of Numbers and Number Systems

The number system is divided into real numbers and complex numbers which are expressions containing imaginary numbers. Real numbers are further classified as rational numbers and irrational numbers. Rational numbers are numbers that can be written as ratios or fractions of two integers. Thus, \(25\), \(-3\), \(\frac{1}{4}\), \(\sqrt4\) and \(0.75\) are rational numbers because:

\[25 = \frac{25}{1}\] \[-3 = -\frac{3}{1}\] \[\sqrt{4} = 2 = \frac{2}{1}\] \[0.75 = \frac{3}{4}\]

and \(\frac{1}{4}\) is a fraction, thus, rational.

Irrational numbers are numbers that cannot be written as fractions or ratios, such as:

• \(\pi\) (pi) \(= 3.1415926\)…

• \(\sqrt{3} = 1.7320508\)…

• Euler’s number = \(2.718281828\)…

Vectors and Matrices

Vectors and matrices are used in solving systems of equations, and the skill in calculating the determinant and performing basic operations is important.

Matrices are described by their number of rows \(M\) and number of columns \(N\), such that a \(3 \times 2\) matrix has \(3\) rows and \(2\) columns.

To calculate the determinant of a matrix with its first row described as \([a, b]\) and second row as \([c, d]\), multiply \(a\) and \(d\) and subtract from this the product of \(b\) and \(c\). To find the determinant of:

\[[3, 5]\] \[[2, 4]\] \[(3 \cdot 4) \;– \;(2 \cdot 5) = 12\; – 10 = 2\]

The method will be different for a \(3 \times 3\) matrix, so it is important to review this, as well as basic operations, such as: addition, subtraction, multiplication, and division that involve vectors or matrices.

Number Theory

A solid foundation in number theory is essential before learning higher math theories. The study of numbers focuses on: natural numbers, integers, the number system, the number line, the properties of even and odd numbers, and other types of numbers. This also includes the relationship between numbers, the concepts of inequality and equality of numbers, ordering, absolute value, prime and composite, factorization, sequences, counting, and more.

Multiples and Factors

Factors are numbers that divide another number, such that the result is a whole number. So \(3\) is a factor of \(9\), and \(4\) is a factor of both \(12\) and \(8\). A related concept is the greatest common factor, or GCF, which is important when solving for the simplest form of a fraction or finding equivalent fractions.

Multiples are numbers that are repetitions of another number, for example, \(9\) is a multiple of \(3\) because \(3 \times 3 = 9\) or it is \(3\) repeated \(3\) times. It is also important to understand the concept of least common multiple, or LCM, of several numbers, especially when adding or subtracting fractions.

Permutations and Combinations

Permutations and combinations are counting techniques. Permutation refers to the number of ways in which numbers, people or things can be grouped when their order is taken into consideration, such as the number of possible 3-digit codes for a lock. The order of the digits in a code is important, so the number of ways can be determined by counting the permutations.

The number of permutations, \(P\), can be solved using these formulas:

When repetition is not allowed:

\[P = \frac{n!}{(n – r)!}\]

Where \(n\) represents the number of things to choose from, and \(r\) represents the number of choices to be made

Or when repetition is allowed:

\[P = n^r\]

Combination, on the other hand, refers to the number of ways in which numbers, people, or things can be grouped when their order is not taken into consideration, such as the number of combinations of shirts you can make if you were to buy only \(10\) from a hundred shirts on sale. Use this formula for computing the number of combinations:

\[C = \frac{n!}{r!(n-r)!}\]

Using “Mental Math” and Reason

It is important to check your answers after computing. The quickest way to judge the correctness of your answer choice is by mentally computing or approximating. Round off numbers, and take note that you may reduce errors in estimating by keeping a balance between rounding up and rounding down. It’s usually safe to estimate when the answer choices given are spaced widely apart.

Use logic and reasoning to validate your answer. When asked about the area of a bathroom, for instance, and you come up with \(200\) sq.ft., you have to inspect your computation to see why a bathroom should be so large. The ability to visualize word problems and question your own answer is a skill that will be very useful for this test, specifically, but your future success as a mathematician or scientist, as well.