# Page 1 - Mathematics Study Guide for the PRAXIS Test

## General Information

*Note: Beginning in September 2019, the math section of this test has undergone changes in the way content is covered and an additional five minutes will be added to test time. All of our materials reflect those changes.*

The Mathematics section of the PRAXIS test is different from the other two sections in the format of its questions. In addition to regular **selected-response** (multiple-choice) questions, you will encounter two additional question formats. One is like the **selected-response** questions, but you will be asked to select **one or more answers**. The other alternative question type requires you to **fill your answer in boxes** provided, with no given choices. This is called a **numeric entry** type of question.

These questions cover a wide variety of skills in these three mathematical domains, in approximately these percentage rates:

**Number and Quantity**(36%)**Algebra and Geometry**[32% (Algebra: 20%; Geometry: 12%)]**Data Interpretation and Representation Statistics, and Probability**(32%)

As opposed to the previous edition of this test, the new PRAXIS® Core Series Math Test has been changed to more **reflect the actual skill needs of all prospective teachers**, regardless of in which field they teach. To accomplish this, test producers engaged the help of over 200 professional educators and made these changes:

- More emphasis on data and statistics
- Fewer questions on algebra and geometry (Previously, algebra and geometry questions occupied 50% of the test. This has been reduced to 32%.)
- Reference sheet with geometric formulas provided (You still need to know
*how*to use them.) -
There are

*no*questions on these concepts:- rational vs. irrational numbers
- solving problems with radicals
- problems working with functions
- problems with 3-D figures (surface area and volume of a cone, etc.)

There is a still great emphasis on solving **real-world problems** on this test, in other words, **word problems**. Many times, the problem-solving process will involve designing a method of combining **several steps** to reach the final answer.

During the PRAXIS® Core Math test, you will have access to a four-function **calculator on the screen**. You will also be given a geometry **formula reference sheet** (new to the 2019 edition of the test). You can see a copy of the formula sheet on page 13 of The Praxis® Study Companion. You will have **90 minutes** to answer **56 questions**.

Following are the concepts covered on this test.

## Number and Quantity

### Integers, Decimals, and Percents

Integers, decimals, and percents are commonly encountered numbers in everyday life. Throw in **fractions** in the form of a numerator over a denominator, and you have covered the world of numbers for most of us except for specialists (like math teachers.)

#### Integers

Take the set of **whole numbers** and combine them with the set of **their negatives**, (except there is no negative zero) and get the set of integers **extending infinitely in both positive and negative directions**.

##### Concepts Concerning Integers

Here’s a reminder, though it’s pretty elementary, that you should know how to **add, subtract, multiply, and divide** positive and negative integers, along with positive and negative decimals and percents. Also elementary is the concept of **even and odd** integers. A couple of additional concepts are listed below.

**Prime Numbers**—If an integer has no factors other than itself and 1, it is a prime number. For example: 13 has no factors other than 1 and 13, so 13 is prime.

**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.

##### 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.

A good way to remember: “Please Excuse My Dear Aunt Sally”

##### Solving Problems with Integers

The properties and operations of integers are likely familiar to you from years of learning arithmetic. As long as you are comfortable with doing **arithmetic using positive and negative integers** you should be fine solving these problems. See below for reminders.

**Adding Integers**—If both integers have the **same sign**, add them and give the result the sign that was common to both.

If the integers have **different signs**, take the absolute value of both and subtract the smaller integer from the larger one. Give the result the original sign of the larger one.

Think \((15)-(8)=7\) and make the \(7\) negative. Result: \(-7\)

**Subtracting Integers**—Change the sign of the second integer and add the resulting integers, following the rules of addition.

**Multiplying And Dividing Integers**— If both integers have the **same sign**, multiply (or divide) and give the result a positive sign. \((-25)\cdot (-6)=150\)

If the integers have **different signs**, multiply (or divide) and give the result a negative sign.

\((14)\cdot (-3)=-42\)

#### Decimals

To be clear, we are talking about **decimal fractions** here, tenths, hundredths, thousandths, and so forth. Decimal fractions tend to be a lot easier to work with than the kind with numerators and denominators, so we see a lot of them in life. Also, they are much **more calculator-friendly**. Although I am calling them decimal fractions here to be as correct as possible, it’s common to just call them **decimals** and that’s what we will do in the rest of this study guide. The word *fractions* from here on will mean the kind with numerators and denominators.

##### All About Decimals

Decimals and fractions also **exist on the number line** that was shown under integers. They actually fill the **space between the integers** (along with the irrational numbers, which we won’t be discussing.)

Understanding decimals is based on understanding **place value**. Starting at the decimal point, each digit to the right represents the next lower fractional division of ten, i.e.

\(\dfrac{1}{10}\) ( 0.1), \(\;\dfrac{1}{100}\) (0.01), \(\;\dfrac{1}{1000}\) (0.001), \(\;\dfrac{1}{10,000}\) (0.0001), etc.

Here is a table showing the place values for \(0\text{.}5124\) (five thousand one hundred twenty-four ten thousandths.) You could argue that the **leading zero** is not really necessary, but it’s good practice to include it to help avoid overlooking the decimal point.

0 | . | 5 | 1 | 2 | 4 |
---|---|---|---|---|---|

x 1 | . | x 0.1 | x 0.01 | x 0.001 | x 0.0001 |

ones | . | tenths | hundredths | thousandths | ten thousandths |

##### Solving Problems with Decimals

A common decimal problem is **converting a decimal to a fraction**. What you do is use place value as a denominator and the decimal digits (without the decimal point) as a numerator and reduce it to the lowest terms. An example is \(0\text{.}20\), which is \(20\) hundredths.

A mixed number example is \(6\text{.}025\), which is \(6\) and \(25\) thousandths.

\[6\frac{25}{1000}=6\frac{1}{40}\]These are easy examples, but the method is the same no matter how unfriendly the numbers are, assuming you don’t run into repeating decimals. It’s unlikely you would see one of those on this test, though you should probably recognize that \(\frac{1}{3}= 0.\bar{3}\) and \(\frac{2}{3}=0.\bar{6}\)

Going from **fractions to decimals** is easier. Just use the **onscreen calculator**, divide, and round if you need to.

Since an onscreen four-function calculator will be available, you will be able to easily add, subtract, multiply, and divide decimals without worrying about the decimal point details that come up when you do these by hand.

#### Fractions

Mathematically, a fraction is simply **a statement of division of two numbers**,

say \(a\) divided by \(b\): \(\dfrac{a}{b}\).

Conceptually, it can be thought of as some entity (maybe a pie) that has been divided into \(b\) parts and there are \(a\) of them being represented. For a non-edible example, each inch is often divided into \(16\) parts on a tape measure, and the diameter of a drill may measure \(7\) of them.

Hence, \(\dfrac{7}{16}\) inch.

##### Things to Know About Fractions

Fractions are written with a top number (**numerator**) and a bottom number (**denominator**). If the top number is bigger than the bottom one, the fraction is called an **improper fraction**. Otherwise, it is a **proper fraction**.

Improper fractions, say \(\dfrac{20}{3}\), can also be written as **mixed numbers**. \(\dfrac{20}{3}= 6\dfrac{2}{3}\). It can be easier to do calculations with the improper fraction than the mixed number, so **changing from mixed to improper** is important. This method works:

##### Solving Problems with Fractions

**Multiplying**—The simplest of the operations, multiplying fractions is done by multiplying the numerators to get a new numerator and multiplying the denominators to get a new denominator. Then reduce it to its lowest terms. For example:

If you are multiplying a mixed number, change it to an improper fraction and do as above.

**Dividing**—If you can multiply fractions, you can divide them. All you need to do is invert the second fraction, then multiply as before. For example:

**Adding**—To add, the fractions must have the **same denominator**. If you can find the **lowest common multiple** of the two denominators, that will simplify reducing after adding. For example:

Multiples of \(6\) are \(12\), \(18\), \(24\), \(30\), and so forth. Multiples of \(10\) are \(20\), \(30\), \(40\), and so forth.

The lowest multiple common to both lists is 30, so that will be the common denominator to use.

**Subtracting**—To subtract, do exactly the same trick to get common denominators, but subtract the numerators instead of adding.

### Ratios and Proportions

A **ratio** shows a **comparison between different quantities**, often written with a colon,

\(3:4\), but may also be written as a fraction, \(\dfrac{3}{4}\).

A **proportion** is a **statement of equality between ratios**. It can be written as

\(3:4=9:12\), but is often written as two fractions: \(\dfrac{3}{4}=\dfrac{9}{12}\).

#### What You Need to Know

Just as two fractions can be **equivalent**, so can two ratios. That’s what a proportion shows (see above.) Also, any ratio can be written as a fraction and vice-versa. For any calculations with ratios, it’s a good idea to write everything in terms of fractions, just because we are more used to working with them.

#### Solving Problems with Ratios and Proportions

Proportion problems usually involve **three given** quantities and **one unknown** quantity. For example, suppose it takes one cup of sugar to make \(24\) molasses cookies. How much sugar will it take to make \(84\) cookies? Set up a proportion using the three given quantities and a variable.

If you add the **units** to the proportion, notice how they work out:

Keep the original numbers (\(1\) and \(24\)) on the left and the new numbers (\(x\) and \(84\)) on the right. Then, if you see the same units in the same positions on both sides, you’ve set it up right. The next step is to multiply both sides by \(84\) to isolate the \(x\).

\[\require{cancel}\] \[84 \times \dfrac{1 \text{ cup}}{24\text{ cookies}}=\cancel{84} \times \dfrac{x \text{ cups}}{\cancel{84} \text{ cookies}}\] \[84 \times \frac{1}{24}=x\] \[x=3.5 \text{ cups}\]