Subtest II: Mathematics Study Guide for the CSET Multiple Subjects Test

General Information

Exactly half of Subtest II of the CSET® Multiple Subjects test is devoted to mathematics. There are a total of 28 questions on math: 26 multiple-choice and 2 that are constructed response, in which you’ll have to write a short, focused answer on your own.

The questions assess your skills and knowledge in a variety of math domains, including numbers, operations, algebra, geometry, measurement, statistics and data analysis, and probability. You will have access to an online calculator for these questions.

Following is a guide to use as you study. Be sure to access additional resources if you encounter problems with any of this content.

Number Sense

Those with good number sense are able to make good judgments about how to use numbers to solve a variety of problems. They can wisely choose what mathematical operations to use and whether the answer they get is reasonable. Sadly, there is no short-cut to achieving this.

Number Ideas

Place Value

Starting at the decimal point, whether it is written or implied, each digit to the left represents the next higher multiple of ten, i.e. 10, 100, 1000, 10,000 etc. Each digit to right represents the next lower fractional division of ten, i.e. 1/10 ( 0.1), 1/100 (0.01), 1/1000 (0.001) etc.

Here is a table showing the place values for 6,374.502

\[\begin{array}{|c|c|c|c|c|c|c|c|} \hline \text{6} & \text{3} & \text{7} & \text{4} & \text{.} & \text{5} & \text{0} & \text{2} \\ \hline \text{thousands} & \text{hundreds} & \text{tens} & \text{ones} & \text{ } & \text{tenths} & \text{hundredths} & \text{thousandths} \\ \hline \end{array}\]

Number Theory

Number theory is the branch of mathematics that deals with interesting patterns in the set of positive whole numbers, known as the natural numbers. There are different ways to group these numbers, some that are well known such as the even numbers, the odd numbers, and the prime numbers, and some that are more curious, such as square numbers and triangular numbers. Listed below are some terms that are important to know.

Greatest common factor—The greatest common factor (GCF) of two (or more) integers is the largest possible integer that can be divided into the two given integers a whole number of times. For example, 12 is the largest integer that can be divided into both 24 and 36.

Least common multiple—The least (lowest) common multiple (LCM) of two (or more) integers is the smallest integer that is a multiple of the two given integers. For example, 18 is the smallest multiple of both 6 and 9.

Prime number—If an integer has no factors other than itself and 1, it is a prime number. For example: 11 has no factors other than 1 and 11, so 11 is prime.

Prime factorization—When an integer is factored in such a way that all factors are prime numbers, that is prime factorization. For example:

\[54 = 2\cdot3\cdot3\cdot3\]

Square numbers—Sometimes called perfect squares, it is worthwhile to know the squares of at least the integers 1 through 12, i.e. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, and 144.

Number Systems

There are infinitely many numbers in mathematics, and to get a handle on them it helps to separate them into groups that have similar properties. Listed below are the names and descriptions of some common groups.

Whole numbers—These are the numbers we all learned in elementary school: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, … and so on, never ending, also called natural numbers, sometimes without 0.

Integers—Take the set of all whole numbers and add them to the set of their negatives, (except there is no negative zero) and get … -4, -3, -2, -1, 0, 1, 2, 3, 4, … the infinite set of integers.

Rational numbers— Any number that can be written as the quotient of two integers is a rational number. For example:

$\frac{1}{3}, \frac{5}{2}, \frac{56}{7}, \frac{-33}{10}$ or even $\frac{6}{1}$

Irrational numbers— Any number that can’t be written as a quotient of two integers is an irrational number. The ones you’re most likely to run into are 𝝅 and roots that don’t equal a whole number. For example:

$\sqrt{2}, \sqrt{35},$ but not $\sqrt{36}$ because that is 6, a whole number.

Real numbers All of the rational numbers and irrationals together make the set of real numbers. For example:

\[-5, \frac{3}{8}, \sqrt{21}\]

Number Order

You need to be able to put numbers in ascending or descending order with any of the following number types or combinations of types. Quick tip: change all numbers to decimal form first.

Whole numbers/integers—The slightly tricky thing about ordering negative integers is that it’s easy to slip up and think -6 is greater than -4, when, in fact, it is less.

Fractions and mixed numbers—The easiest way to order these is to first use a calculator to change all numbers to decimal form. Ordering should then be pretty obvious.

Other rational numbers—Similarly, to put percents in order with other numbers, first change the percents to decimal form.

Irrational numbers—Although irrational numbers can’t be expressed exactly in decimal form, a calculator will give you an approximate decimal form with enough accuracy to use.

Special Notation

To deal with very large or very tiny numbers, scientific notation, also called exponential notation, has been developed. Be able to perform operations on these.

Scientific notation: addition—This notation is written in the form $a \times 10^{r}$ where a is a number between 1 and 10 and r is an integer.

You have to adjust the decimal points and r exponents to get both numbers to have the same r values. Then add the a values together and write $\times\; 10^{r}$.

For example:

\[1.2 \times 10^{3} + 5.1 \times 10^{4} = 0.12 \times 10^{4} + 5.1 \times 10^{4}\] \[(0.12+5.1)\times 10^{4} = 5.22\times10^{4}\]

Scientific notation: multiplication—To multiply two numbers in scientific notation, multiply the a parts and add the r parts.Then adjust the decimal point as needed.

For example:

\[(2.3 \times 10^{5}) (7.42 \times 10^{-3}) = (2.3\cdot7.42) \times 10^{5+(-3)} = 17.066 \times 10^{2}\]

To make the a part between 1 and 10, divide 17.066 by 10 and then to compensate for that, add 1 to the r part.

This gives:

\[1.7066 \times 10^{3}\]

Working with Exponents

Exponents are often positive integers, but can also be negative, and may be positive or negative fractions as well.

Negative exponents—A negative exponent is handled by taking the reciprocal of the base and rewriting its exponent as positive. For example:

\[3^{-2} = \frac {1}{3^{2}}\]

Fractional exponents—A fractional exponent, say $a^{\frac{x}{y}}$ is handled by rewriting it as a y root of a, raised to the x power, like this:

\[a^{\frac{x}{y}}=(\sqrt[y]{a})^{x}\]

Exponents of fractions—Just as integers can be raised to a power, so can fractions.

\[\bigg(\frac{2}{5}\bigg)^{3} = \frac{2}{5} \cdot \frac{2}{5} \cdot \frac{2}{5} = \frac{8}{125}\]

Operations with Positive and Negative Numbers

Operations with positive and negative numbers such as addition, subtraction, multiplication, and division have exact rules you need to know to deal with the signs.

Relationships between Operations

Each math operation can be thought of as having an opposite operation that “undoes” it: Subtraction is the opposite of addition; Division is the opposite of multiplication; Taking the square root is the opposite of squaring, to name three examples.

Properties

There are certain properties of number systems that may hold true for some operations and not others. Listed below are some of the common ones.

Associative property— For multiplication and addition, how you group (associate) the numbers doesn’t matter. For example:

\[(3 + 9) + 5 = 3 + (9 + 5)\]

Commutative property—For multiplication and addition, the order of the numbers doesn’t matter. For example:

$4 \cdot 6 = 6\cdot 4$ and $4 + 6 = 6 + 4$

Identity property— 0 added to a second number doesn’t change that number’s identity. Likewise, 1 times a second number doesn’t change its identity. 0 is the additive identity and 1 is the multiplicative identity.

Distributive property— $a(b + c) = ab + ac$ Think of a as being distributed to the b first and then c.

Working with Numbers

Here are a few general things to keep in mind when working with numbers.

Algorithms

An algorithm is just a series of steps that will solve a certain problem. Remember how to add two two digit numbers? 1. Add the ones column. 2. If the result is over ten, carry a one to the tens column, and so on. That procedure is an algorithm. Standard algorithms exist for adding, subtracting, multiplying, and dividing whole numbers, fractions, and decimals, to name the most common ones that you should know. It’s often said that there’s more than one way to skin a cat, (ew!) and you may see an algorithm that you haven’t seen before. Usually you can use one you do know to check the correctness of the new one.

The Order of Operations

Remember to follow the order of operations (PEMDAS):

Do everything in parentheses (P), left to right.
Evaluate any exponents (E), left to right.
Do all multiplication and division MD), in order, left to right.
Then do all addition and subtraction (AS), in order, from left to right.

Good way to remember: Please Excuse My Dear Aunt Sally

Rounding and Estimating

Rounding is a way of changing a number to a more approximate version of itself. If you knew that something cost $158.95, you would very likely just think of it as $160 (rounding to the nearest ten dollars).

The simplest rule of rounding is to look at the place to which you want to round and, if the digit to the right of it is 5 or greater, add 1 to the targeted place and drop, or turn into zeros, all non-zero digits to its right.

For example: Round 409 to the nearest ten. To the right of the tens place is a 9 (bigger than 5), so add 1 to the tens place and change the 9 to a zero, giving 410. You should review rounding decimals as well.

Estimating means to use rounded numbers to quickly get an approximate answer. For example: Estimate the product of $53 \times 288$. Round 53 to 50 and 288 to 300.

\[50 \times 300=15,000\]

Using Technology for Complex Calculations

Scientific and graphing calculators are common, and they make it easier to work with scientific notation, probability, and other calculations. Spreadsheets can help deal with a lot of data, as can a computer program that you write.