# FTCE General Knowledge: Ultimate Guide and Practice Test

Preparing to take the FTCE General Knowledge test?

Awesome!

You’ve found the right FTCE General Knowledge study guide. We will answer every question you have and tell you exactly what you need to study to pass the FTCE General Knowledge test.

In fact, we will cover 4 key areas of the exam.

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# FTCE General Knowledge Videos

# FTCE General Knowledge Overview

FTCE General Knowledge may be a required exam for anyone seeking teacher certification in any grade in Florida. It is a test of basic skills. The test ensures that teachers at all levels possess the necessary competencies and skills to effectively teach.

There are 4 subtests:

- Essay Subtest
- English Language Skills Subtest
- Reading Subtest
- Mathematics Subtest

You can register to take all or any combo of the four subtests in one session. You have to pass all subtests to earn a passing score for the exam.

The exam is a computer-based test, but don’t worry, you just need basic computer skills. If you made it to this point in your education and career, you have nothing to worry about.

Here is the subtest format and time allotment breakdown:

Subtest | Questions | Time |
---|---|---|

Essay | 1 Essay | 50 Minutes |

English Language Skills | 40 Multiple-Choice | 40 Minutes |

Reading | 40 Multiple-Choice | 55 Minutes |

Mathematics | 45 Multiple-Choice | 100 Minutes |

## Frequently Asked Questions About the FTCE General Knowledge

### Quick Facts

**Cost: **$32.50 (one subtest); $65 (two subtests); $97.50 (three subtests); $130 (four subtests)

**Location: **Tests are by appointment only, year-round. Click here to find a testing site near you.

**What to Bring: **You need to bring two valid, unexpired forms of identification in English. Click here for more information about identification requirements.

**Number of Attempts: **There are no limits on how many times you can take the test before you pass; however, you do have to wait 31 days between testing attempts.

**Scoring**: You must score at least 8 out of 12 points on the Essay Subtest to pass. For the English Language Skills, Reading, and Mathematics Subtests, you must score at least 200 points on each to pass.

When you finish taking the test, you will immediately get an unofficial pass/non-pass result based on the multiple-choice questions (unless they are experiencing score report delays). You can expect your official score within four weeks.

Click here to read more about testing policies.

Register for the FTCE General Knowledge exam here.

### Frequently Asked Questions

**What FTCE test(s) do I need to take?**

The Bureau of Educator Certification (BEC) determines your testing requirements after you apply for certification. Click here for more info.

You may be able to demonstrate mastery of General Knowledge in another way. Click here for more info.

**How long will the FTCE General Knowledge test take me?**** **

The full test itself is 245 minutes but expect to be at the testing site longer. It takes time to get checked in and get started. Plan to arrive at least 30 minutes before your appointment time.

**Do I get a break during the FTCE General Knowledge exam?**

If you take all four General Knowledge subtests in a single session, you will receive a 15-minute break.

**What can I expect when I arrive at the testing site to take the FTCE General Knowledge test?**** **

When you get there, expect to have your identification checked, your photo taken, your palm scanned, your eyeglasses checked, and to place your belongings in secure storage. You will be given an erasable notepad and pen. Before starting the test, you’ll have to complete a short tutorial and sign an agreement.

**How do I pass the FTCE General Knowledge test?**

To pass the FTCE General Knowledge test, you must first understand what is on the exam and what you will be expected to know. The best way is to review the 240Tutoring test breakdown materials and practice questions. Once you identify areas of weakness, you can begin targeting those areas with instructional content and practice questions.

**Is the FTCE General Knowledge test hard?**

Yea, it’s challenging. Take a look at these passing rates:

Source: Florida Department of Education, Division of Accountability, Research, and Measurement, Office of Assessment, Postsecondary. Data are provided by Evaluation Systems group of Pearson February 2018.

You have to study quality, trusted sources (like 240Tutoring).

## FTCE General Knowledge 5 Top Tips

#### Make sure to answer every question (even if you guess)

#### Know how much time you have left

#### Eliminate incorrect answers first

#### Work through practice questions so you know what to expect

#### Study quality, trusted sources (like 240Tutoring)

## FTCE General Knowledge: Essay

### Overview

You will have 50 minutes to complete 1 essay question. You must prepare, write, and edit your essay within the 50 minutes. A whiteboard will be given to you so you can plan your essay.

Two raters will score your essay based on content and these elements:

- ideas
- focus
- organization
- style
- diction
- sentence structure
- mechanics
- capitalization
- punctuation
- spelling
- usage

Take a look at the official skills assessed via the essay:

Check out the 240Tutoring Constructed Response Questions page. It has some very good information that will help you on the test.

## FTCE General Knowledge: English Language Skills

### Overview

You will have 40 minutes to complete 40 multiple-choice questions.

The English Language Skills subtest can be neatly divided into three different sections. Those three sections are:

- Language Structure
- Vocabulary Application
- Standard English Conventions

So, let’s start with Language Structure.

### Language Structure

This section tests your knowledge on how the English language works (or should work).

Language Structure questions make up about 25% of the English Language Skills subtest.

There are four big concepts you definitely have to know to get these questions correct:

- Modifiers
- Parallelism
- Effective Structures
- Patterns of Organization

**The first big concept you should know is what a modifier is**. Modifiers are words or phrases that change or clarify another word in the sentence. The purpose of a modifier is to add detail or explain something. These words are usually adjectives (describe nouns) or adverbs (describe verbs).

Here is an example. Look at this simple sentence:

They were running.

Now, let’s add a modifier:

They were running quickly.

The adverb *quickly *is a modifier of the verb *running*.

Get it? Good!

But, modifiers are commonly misplaced. Look at this sentence:

She purchased a car for my brother they call Lightning.

In this sentence, Lightning is the car’s name. But the placement of the modifier is all wrong, so it causes some confusion. Is Lightning the brother’s name? No! Let’s move the modifier closer to the noun it modifies:

She purchased a car they call Lightning for my brother.

See? All better.

Here’s a friendly tip: These are very simple sentences I just used. Expect the sentences on the test to be much more detailed.

**The next big concept to know is parallelism. **Very simply, parallel structure means that you use the same grammatical form within a sentence.

Let’s look at an example of a sentence __without__ parallel structure:

Caroline enjoys dancing, the playground, and to take long walks.

Now, look at the same sentence but __with__ parallel structure:

Caroline enjoys danc__ing__, go__ing__ to the playground, and tak__ing__ long walks.

See how all the verbs end in –*ing*? That’s parallelism!

Again, these are simple sentences. Expect to see more detailed, complex sentences on the exam.

**The third big concept to know is effective structures. **This includes knowing how to identify and fix:

- fragments
- comma splices
- run-on sentences
- syntax errors

I’ll talk about a couple of these in more detail in a little bit.

**And the last big concept to know is patterns of organization. **More specifically, you need to know what the modes of rhetoric are. Here, let me tell you. The most common modes are:

- narrative
- descriptive
- expositive
- argumentative

Now, those are the four broad concepts to be familiar with.

**Specific Concepts**

Right now, I’m going to give you two specific concepts to be familiar with because they will most likely appear on the test.

**Comma Splices**

Comma mistakes happen. They happen a lot. A comma splice is a common comma error (wow, that was a mouthful). This error occurs when two independent clauses are joined together with a comma. It is a type of run-on sentence. Let’s look at an example:

Tomatoes are not vegetables, they are a fruit.

These two independent clauses can be sentences on their own, so simply adding a comma between them just doesn’t work.

We can fix this comma splice error three ways:

By adding a conjunction:

Tomatoes are not vegetables, *but* they are a fruit.

By changing the comma to a semicolon:

Tomatoes are not vegetables; they are a fruit.

By making separate sentences:

Tomatoes are not vegetables. They are a fruit.

You can expect to have to correct a comma splice error on the test.

**Sentence Fragments**

A sentence fragment is a group of words that look like a sentence but aren’t. A sentence needs at least one independent clause, containing a subject and a verb. Here is an example of a fragment:

Bought the groceries.

This fragment begins with a capital letter and ends with a period, but it is not a complete sentence. Where is the subject? Who bought the groceries? Let’s fix it:

*Jonathan *bought the groceries.

There you go. Now it’s a complete sentence with a subject and a verb (independent clause).

You’ll more than likely need to identify and fix sentence fragments on the exam.

### Vocabulary Application

Put simply, this section tests your knowledge about words.

Vocabulary Application questions make up about 25% of the English Language Skills subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Determine Meaning of Words
- Correct Usage of Words and Phrases
- Diction and Tone

**The first big concept you should know is how to use context to figure out what a word means. **You will come across a word that you probably don’t know on the test. By using the context (the sentences around the word) and clues, you should be able to figure out what that word means.

Parts of words, like prefixes, roots, and suffixes, can really help you figure out what a word means.

**The next big concept to know is how to correctly use words and phrases**. On the test, you will definitely be given a sentence with a word missing. You will have to choose the most effective word or phrase within the context to place in the sentence.

**And the last big concept to know is what diction and tone are and how to identify them in writing. **Diction refers to the author’s choice and use of words and tone is the way the author expresses his or her feelings and attitudes through those words.

Those are the three broad concepts to be familiar with.

**Specific Concept**

Let’s talk about a specific concept that you need to be familiar with.

**Commonly Misused Words**

There are certain words that people just have a hard time with. Take a look at this list:

- affect/effect
- lie/lay
- you’re/your
- imply/infer
- comprise/compose
- farther/further
- fewer/less
- cite/site
- than/then
- to/too/two
- there/their/they’re

For each of the bullets above, make sure to know when the right time to use each word is. You will more than likely see at least one of them on the test.

**Prefixes, Roots, and Suffixes**** **

You can use a knowledge of prefixes, roots, and suffixes to figure out what a word means.

For example, let’s look at the word *portable *(I know, you probably already know what this word means, but it’s a good example). If you have a good understanding of prefixes and suffixes, you can gather a good idea about what *portable *means.

The prefix *port *means *to carry *and the suffix *able *means *capable of*. So, put them together and you can figure out that the word *portable *means *capable of being carried *or something like that.

Be familiar with the most common prefixes, roots, and suffixes. Check out this resource. It will help you on the test.

### Standard English Conventions

This section tests your ability to identify and correct errors in sentences. These include tense, subject/verb agreement, parts of speech, spelling, punctuation, and capitalization errors.

Standard English Conventions questions make up about 50% of the English Language Skills subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Parts of Speech
- Capitalization, Punctuation, and Spelling Conventions
- Agreement

**The first big concept to know is the parts of speech. **You definitely need to know what these are:

- noun
- pronoun
- verb
- adverb
- adjective

Be sure to know what these parts of speech are and how to correctly use them in sentences.

**The next big concept to know is standard capitalization, punctuation, and spelling conventions. **These are the rules of capitalizing, punctuating, and spelling. Take some time to review all of those rules you learned in grade school.

Remember, on the test, you’ll be finding errors in sentences.

**And the last big concept is agreement. **There are two types of agreement that will appear on the exam: subject/verb and pronoun/antecedent. Know what both of these are and how to fix sentences with agreement problems.

Now, those are the three broad concepts to be familiar with.

**Specific Concepts**

Right now, I’m going to give you two specific concepts to be familiar with.

**Pronoun Shifts**

This is a common grammatical error that occurs when a writer starts a sentence using one type of pronoun and then suddenly shifts to another type of pronoun.

Let’s look at an example:

If __you__ brush __your__ teeth regularly, then most __people__ can avoid cavities.

This sentence starts with the pronouns *you* and *your* (singular) but then suddenly shifts to *people* (plural). This is an error. Let’s fix it:

If __you__ brush __your__ teeth regularly, then __you__ can avoid cavities.

See how it all matches now? No sudden shifts!

**Subject-Verb Agreement**

In sentences, subjects and verbs must agree in number. This means that if a subject is singular, its verb also needs to be singular. If the subject is plural, then its verb must also be plural. Let’s look at a sentence with incorrect subject-verb agreement:

Emily are at the pool.

The subject, *Emily*, is singular. Emily is one girl. The verb, *are*, is plural. Let’s fix this sentence:

Emily *is* at the pool.

Now, the subject and verb are both in singular form. They agree.

The trick to making sure your subjects and verbs agree is knowing the singular and plural forms of subjects and verbs. There are some rules that are helpful to know.

And that’s some basic info about the English Language Skills subtest.

Now, let’s look at a few practice questions in each area to see how these concepts might actually appear on the real test.

## FTCE General Knowledge: Reading

### Overview

You will have 55 minutes to complete 40 multiple-choice questions.

All Reading questions are passage based. Reading passages will be both expository and narrative. Each test form will contain about five passages.

The Reading subtest can be neatly divided into three different sections. Those three sections are:

- Key Ideas and Details
- Craft and Structure
- Integration of Information and Ideas

So, let’s start with the biggest, Key Ideas and Details.

### Kay Ideas and Details

This section tests your knowledge on drawing inferences and conclusions, identifying themes, and determining the cause and effect of events.

Key Ideas and Details questions make up about 40% of the Reading subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Conclusions and Inferences
- Cause and Effect
- Central Idea and Themes

**The first big concept is knowing what conclusions and inferences are and how they differ from each other. **

A conclusion is a judgement or decision reached based on what you learned from the passage.

An inference is figuring out missing information by using clues and hints from the passage. When trying to make an inference, a good question to ask yourself is, “What does the passage suggest?” An inference is a *type *of conclusion.

Make sure your inferences rely on the author’s words rather than your own feelings or experiences.

You can 100% expect to be asked to draw a conclusion or make an inference after reading a passage.

**The next big concept is understanding what cause and effect is. **A cause/effect relationship describes or discusses an event or action that is caused by another event or action. This structure often uses words and phrases such as because, since, as a result, due to, consequently, and therefore.

Remember, one cause (or event) can have several effects. For example, crashing your car can cause broken bones, increased insurance premiums, and angry parents. This one event had three (negative) effects.

You can expect to see cause and effect questions on the test.

**And the last big concept to know is what central ideas and themes are and how to identify them. **Think of main ideas and central ideas as the same thing. They are what the story is mostly about. Main ideas are usually found in literary passages while central ideas are usually found in informational text.

The theme of a piece of text is the message or lesson the author wants to get across. It is usually a statement about life that readers can apply to their own lives.

You also need to know *how *to identify the central/main ideas and themes of reading passages. This will definitely be on the test. A great resource to have would be a very thorough study guide (I know of a great one!).

Now, those are the three broad concepts to be familiar with.** **

**Specific Concepts**

Right now, I’m going to give you one specific concept that will be on the test.

**Supporting Information/Evidence**

The phrase “textual evidence” comes up several times in the Reading subtest competencies and skills.

Reading passages contain supporting information, or evidence, which includes facts, statements, or examples that guide you to a full understanding of the main idea. This evidence is used to clarify, illuminate, explain, describe, expand, and illustrate the main idea.

Sometimes a test question will ask you to choose which supporting evidence from a passage best proves the author’s claim.

Some examples of supporting information/evidence include:

- comparison- one thing is shown to be like another
- contrast- one thing is shown to differ from another
- statistics- use of research to support an observation
- quotations- use of the words of authorities to add weight to an argument
- vivid descriptions

### Craft and Structure

This section tests your knowledge on how well you can identify and explain an author’s word choice, organization, point of view, and purpose.

Craft and Structure questions make up about 25% of the Reading subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Word Choice and Meaning
- Organization and Text Structure
- Point of View and Purpose

**The first big concept to know is word choice and meaning**. Why did the author choose to use this word and what does it mean? Specific word choices can shape a text’s meaning or tone. Author’s carefully choose their words when writing a story. Authors may choose to use different types of language like figurative or connotative language.

For the test, you need to be able to figure out the meaning of words and phrases and how those specific word choices affect the meaning or tone of the passage.

**The next big concept to know is about story organization and text structure. **For the test, think of these words as synonyms. Organization is the act of putting things into a logical order, and can refer to writing, thoughts, or even your sock drawer!

In writing, organization is the arrangement of ideas, incidents, evidence, or details in a specific order. Writers use different organizational strategies, and you need to be able to recognize a few of the most utilized methods of organization like cause/effect, compare/contrast, and problem/solution.

**And the last big concept is point of view and purpose. **You need to be able to figure out *how *an author views something and *why *an author wrote the passage. On the test, you may be asked to compare the point of view of two or more authors on the same topic.

Know these purposes for writing:

- to persuade
- to inform
- to entertain
- to compare
- to express

Now, those are the three broad concepts to be familiar with.

**Specific Concepts**

Right now, I’m going to give you two specific concepts to be familiar with because they will most likely appear on the test.

**Figurative Language**** **

Figurative language is language that uses words with meanings that are different from the literal interpretations. A writer using literal language simply states the facts as they are. A writer who describes something by comparing it with something else is using figurative language.

Writers use figurative language to be more effective and persuasive and to give the readers new insights. The most common figurative language are similes, metaphors, and personification.

**Connotative Language**

The emotion and association connected to a word is known as its connotative meaning. Since different people have different experiences, words can connotatively mean different things. Words can have a negative, neutral, or positive connotation.

A word’s dictionary definition is known as its denotative meaning.

### Integration of Information and Ideas

This section tests your ability to evaluate, synthesize, and analyze information from texts.

Integration of Information and Ideas questions make up about 35% of the Reading subtest

There are really only three things you need to know how to do for this part of the test. You need to know how to:

- Evaluate a statement
- Synthesize information
- Compare two statements

**The first thing you need to know how to do is evaluate an author’s statement for relevancy, sufficiency, and validity**. The test may ask you to read an author’s statement from a passage and then determine if that statement is valid or invalid and why.

**The next thing you need to know how to do is synthesize information**. Basically, you need to take all of your background knowledge (the stuff you already know) and what you learn from a reading passage to form a better understanding of what the author is trying to say.

**And finally, you need to be ready to compare two statements from a passage**. On the test, you will for sure be asked to read two sentences from a passage and determine if the second sentence contradicts, explains, restates, clarifies, describes, or relates the first sentence.

And that’s some basic info about the Reading subtest.

Now, let’s look at a few practice questions in each area to see how these concepts might actually appear on the real test.

## FTCE General Knowledge: Mathematics

### Overview

You will have 100 minutes to complete 45 multiple-choice questions.

Reminder: The test center will provide an on-screen four-function calculator and an on-screen reference sheet.

The Mathematics subtest can be neatly divided into four different sections. Those five sections are:

- Number Sense
- Geometry and Measurement
- Algebraic Thinking
- Probability and Data

So, let’s start with Number Sense.

### Number Sense

This section tests your ability to order numbers and correctly solve real-world problems using the four operations.

Number Sense questions make up about 17% of the Mathematics subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Order of Operations
- Value of Real Numbers
- Word Problems

**The first big concept and by far the most important is order of operations**.

The order of operations, or PEMDAS, P-E-MD-AS, is simply the process you follow to simplify and work an equation.

Let me give you an example. If you have an equation like this:

2(x – 3) + 3(x + 4)^{2}

You have to work the problem according to a specific order, the order of operations.

In this particular example, you would first solve for the parentheses, then you would solve for the exponents. Then going left to right, you do either multiplication or division then you would add or subtract in the same order (from left to right).

So that’s why we group the M and the D and the A and the S together.

Now, this is incredibly important because, on the test, you will be required to simplify an equation. So, you have to know the order of operations to get that question correct.

**The next big concept to know is the value of real numbers**. A real number is, really, any number you can think of. It includes whole numbers, rational numbers (fractions and decimals), and irrational numbers (π, √2, etc.). Real numbers can be negative or positive.

Now, the test will, and I mean will have you place different forms of numbers from greatest to least or least to greatest on a number line.

In a given data set, you might have two fractions, a decimal, pi, a negative integer, and a regular integer. You then must organize all of these from least to greatest (or greatest to least).

So, make sure you know how to translate decimals to fractions and fractions to decimals, so you can compare the two to figure out which one is greater or less than the others.

Oh, and also make sure you understand decimals to the hundredths place.

**Now, the last major concept to know from number concepts is how to read, structure, and apply mathematical word problems. **These problems will include all four operations (+, -, x, ÷).

Now, my biggest tip to help you with this is simply to work through a lot of authentic practice questions, specifically, word problems in mathematics.

Now, while these questions aren’t going to be the most complex, it does take a lot of practice to learn how to read a question prompt, understand the question, and then solve the problem.

And really, it’s just one of those things you have to practice, practice, practice. So, find a great source of authentic practice questions that you can use.

Those are the three broad concepts to be familiar with.

**Specific Concepts**

Right now, I’m going to give you two specific concepts to be familiar with because they will most likely appear on the test.

**Prime Numbers**** **

A prime number is a whole number that cannot be made by multiplying other whole numbers together. Examples of prime numbers include 2, 3, 5, 7, and 11. There are many more.

You should also know what prime factors are and how to find them.

**Properties of Math**

Think of a property as kind of like a rule you need to follow. Okay? There are quite a few of these properties:

- Commutative Property of Addition
- Commutative Property of Multiplication
- Associative Property of Addition
- Associative Property of Multiplication
- Distributive Property
- Additive Identity Property
- Additive Inverse Property
- Multiplicative Identity Property
- Multiplicative Inverse Property

Whew. Like I said, there’s a lot. You need to know what these are and how to solve problems using them. A really thorough study guide would be a great resource to have and use.

### Geometry and Measurement

This section tests your ability to identify and classify shapes and solve measurement problems.

Geometry and Measurement questions make up about 21% of the Mathematics subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Two- and Three-Dimensional Shapes
- Ratio and Proportion
- Measurement Units and Quantities

**The first big concept to know is how to identify and classify both two- and three-dimensional shapes based on their properties. **Properties of two-dimensional shapes can include sides, corners, and angles.

Properties of three-dimensional shapes can include faces, edges, and vertices.

You also need to know how to find the perimeter and area of two-dimensional shapes and the surface area and volume of three-dimensional shapes.

**The next big concept to know is what ratio and proportion are**. On the test, you will be asked to solve ratio and proportion problems by looking at scaled drawings. Let’s talk about what these three words mean.

A **ratio **simply compares numbers. A ratio can be written in three different ways:

3:1

3 to 1

3/1

A **proportion **means that two ratios are equal. For example, 1/4 = 2/8. These ratios are equal because if we reduce 2/8, we arrive at 1/4.

A **scaled drawing **is a drawing that represents a really large object (too large to be drawn in its actual dimensions).

So, know these terms and do several practice problems to get ready for the test.

**And the last big concept to know is measurement units and quantities. **

You need to be familiar with units and quantities like:

- temperature
- time
- money
- length
- width
- height
- area
- mass
- weight
- volume
- speed

Now, just like with word problems and number concepts, this mathematical concept isn’t necessarily complex, you just need to make sure you’re familiar with those different types of measuring units and quantities.

Those are the three broad concepts to be familiar with.

**Specific Concepts**

**Pythagorean Theorem **

The Pythagorean Theorem is used to find any length of a side of a right triangle (if you know the other two sides).

The equation of the Pythagorean Theorem is:

a^{2 }+ b^{2}= c^{2}

Where A and B are the two sides of the right triangle and C is the hypotenuse of the triangle.

And I guarantee you one thing, if you take anything from this, take this one thing: you will have a question about the Pythagorean Theorem on the test. That is an absolute guarantee. It will most likely be in some sort of word problem.

An example would be:

Billy walked three blocks west, and four blocks north. If Billy walked a straight line, how many blocks would Billy have walked?

Something like that will show up on the test.

**Area**

On the test, you will be asked to find the area of two-dimensional shapes and the surface area of three-dimensional shapes.

But don’t worry! Remember, I told you that you’ll be given an on-screen reference sheet for the Mathematics subtest. This reference sheet lists all of the formulas for finding area and surface area.

BUT, you do need to spend some time practicing these types of problems. It won’t be as easy as just finding the area of a rectangle (length x width). You might be asked to find the area of a trapezoid or the surface area of a cylinder.

And you can bet these skills will be tested with real-world word problems.

### Algebraic Thinking

This section tests your ability to, you guessed it, solve algebra problems.

Algebraic Thinking questions make up about 29% of the Mathematics subtest.

There are three big concepts you definitely have to know to get these questions correct:

- Solving for x
- Slope-intercept form
- Creating an equation from a data set

**The first big concept you need to know is how to solve for x. **Solving for x is pretty straightforward. The test will give you an equation and then you have to solve for x. So, the test will give you an equation like:

4x^{2}– 4(3 + 2) = 16

And in this equation, we would just need to balance the equation for x, so you would simplify it as much as you can, balance the equation, and you’d find that x equals three.

4x^{2 }– 4(5) = 16

4x^{2 }– 20 = 16

4x^{2}= 36

x^{2}= 9

x = 3

Now, take note. You are more than likely going to be presented with a real-world word problem. Be able to take what you need from the problem, build the equation, and solve for x.

You are also going to have to solve inequalities. Solving inequality statements in one variable, such as 3/4x – 9 > 21, is much the same as solving equations like those addressed above.

The only major differences are that:

- the direction of the inequality sign will change after some operations are performed
- a finished final answer generally requires the variable to be written on the left
- the solution set is often presented on a number line.

**And the second concept is slope-intercept form.**

Now, the slope-intercept equation is simply:

y = mx + b

And in this equation, m is equal to the slope of the line on a graph and b is the y-intercept, hence the name slope-intercept form.

So, for the equation:

y = 3x + 4

The slope of the line would be three and the y-intercept would be four.

**Now, the last major concept to know is how to create an equation from a data set. **The test will give you a data set that looks something like this:

x | 1 | 2 | 3 | 4 |

y | -1 | -3 | -5 | -7 |

And you’re required to create a corresponding equation that matches the data set. For this data set, the corresponding equation is:

y = 1 – 2x

Now, if you look at the data set, anytime you plug the x value in, you get the corresponding y value if you solve for the equation. And while this kind of question can seem difficult, it’s one of the easier questions to answer if you just work backward.

All you need to do is look at the answer options, plug in the data set values into each equation, and see if they match.

Now, those are the three broad concepts to be familiar with.

**Specific Concepts**

**Distributive Property of Multiplication over Addition**** **

There are a lot of properties (think: rules) in algebra. Knowing them will help you solve algebra problems quickly and correctly.

Take a look at one of these properties, the Distributive Property of Multiplication over Addition:

*The product of a factor with a sum is the same as the sum of the products made by that Algebra factor and each number involved in the addition or subtraction. The reverse direction is also true. *

Symbolically, then, the Distributive Property can be represented multiple ways:

a(b + c) = ab + ac

(b + c)a = ba + ca

ab + ac = a(b + c)

ba + ca = (b + c)a

As subtraction is properly understood as a form of addition, each statement above is also true for subtraction, such as a(b – c) = ab – ac.** **

**FOIL**

Remember the acronym FOIL from middle school algebra? The letters of

FOIL stands for the pairs of values to be multiplied during the distribution process before being added together.

For example, in the problem (a + b)(c + d):

F = First, the first quantity from each binomial = ac

O = Outer, terms towards the outside of the binomial multiplication problem = ad

I = Inner, the terms towards the inside of the binomial multiplication problem = bc

L = Last, the last term in each binomial = bd

And so, the product of (a + b)(c + d), obtained through the Distributive Property using FOIL, is the sum of the terms ac + ad + bc + bd.

### Probability and Data

This section tests your ability to determine the likelihood of events occurring and analyze and interpret sets of data.

Probability and Data questions make up about 33% of the Mathematics subtest. It’s a big one.

There are three big concepts you definitely have to know to get these questions correct:

- Measures of Central Tendency
- Organizing and Displaying Data
- Probability

**The first big concept to know is measures of central tendency. **What does that mean?

It’s simply four concepts: mode, median, mean, and range.

Now, on the test, they’re going to give you a data set of about 8 to 12 numbers, and they’re going to ask you one or more of the following. What is the mode? What is the median? What is the mean? Or what is the range of the data set? They will also be embedded in real-world word problems.

So, you have to know what those central tendency measurements are and how to find them for a data set. And when you practice, work on a data set of about 8 to 12 double-digit numbers.

**The next big concept to know is the different ways to display and organize data. **On the test, you will see data displayed in the following ways:

- histograms
- bar graphs
- circle graphs
- pictographs
- line plots
- tables

Know what each of these are and how the presentation of data can lead to different or inappropriate interpretations in the context of a real-world situation.

**And, finally, the last big concept to know is probability. **Probability is defined as the likelihood of an event occurring. Probability is the number of successful outcomes possible to the total number of outcomes possible.

Look at these examples.

If a six-sided die is rolled, what is the probability of landing on any one of the sides? The answer, of course, is one in six.

The next question you might get is: Billy flipped a coin 10 times. The coin landed on heads seven times, and the coin landed on tails three times. What is the probability of the coin landing on tails on the next flip?

You also need to know how to calculate probability using counting procedures, tables, and tree diagrams.

Now, those are the three broad concepts to be familiar with.

**Specific Concepts**

**Standard Deviation**

You need to what standard deviation is. The standard deviation of a data set is found by taking the (positive) square root of its variance. When the measures of central tendency are closely clustered so that the mean is approximately equal to the median, which is also approximately equal to the mode, and the data is dispersed symmetrically about the median, this is called a normal distribution and is often referred to as a bell curve. In a normal (bell-shaped) distribution, about two-thirds of the data lie between ±1 standard deviation of the mean.

Did that make your head hurt? Well, put more simply, standard deviation is a measure of how spread out numbers are.

**Odds**** **

So, people often use the terms “odds” and “probability” interchangeably. But, there is an important difference between the two.

Probability = the number of successful outcomes possible: total number of outcomes possible.

Odds = number of successful outcomes possible: number of unsuccessful outcomes

Know the difference between the two. It’s important.

And that’s some basic info about the Mathematics subtest.