# Next-Generation ACCUPLACER

## Preparing to take the ACCUPLACER?

Awesome!

You’ve found the right page. We will answer every question you have and tell you exactly what you need to study to score well on the ACCUPLACER.

Reading Placement Test

Writing Placement Test

Mathematics: Arithmetic

Mathematics: Quantitative

Mathematics: Advanced

## Quick Facts

Get the “need to know” information at a quick glance.

**Overview**

The ACCUPLACER is a web-based assessment that measures student readiness for college courses. This test focuses on the knowledge and skills that research shows to be essential for college readiness and success. Your ACCUPLACER test score will help colleges, including universities and technical schools, make placement decisions and determine where you will be likely to succeed. The ACCUPLACER focuses on three major areas of content knowledge: Reading, Writing, and Math.

**Format**

Let’s take a look at how you can expect the ACCUPLACER to be formatted:

Feeling overwhelmed by the number of questions on the ACCUPLACER? Don’t be – that’s because the ACCUPLACER is untimed!

It’s also important to know that you cannot return to previous questions on the ACCUPLACER. This is one reason why you should take advantage of the unlimited time available to you as you take the test.

The ACCUPLACER is a computer adaptive test (CAT). This means that the computer will assign the next question or question set based on the response you gave to the previous question. In other words, this test is designed to fit your ability level.

**Scoring**

One aspect of the ACCUPLACER (which students really appreciate) is that you cannot fail the test. This is because the ACCUPLACER is not designed to measure what you do or do not know; the test is designed to measure what you are *ready* to learn. Think of it this way, ACCUPLACER means “accurate placer” – in other words, it focuses on placing you where you will succeed. It does not focus on your prior learning.

Although you cannot pass or fail the ACCUPLACER, you will receive a score after taking the exam. Scores on the ACCUPLACER range from 200 – 300. You will receive your score as soon as you finish the exam. No wait, no sweat!

**Study Time**

Studying appropriately for the ACCUPLACER is going to be your best bet when it comes to putting yourself in a position for future success. Consider this – if you do not score your best on the ACCUPLACER, you may be placed in a remedial class that you do not need.

The amount of time you will need to spend studying for the ACCUPLACER exam depends upon your existing skills and how much information you have retained from previous courses. You can use Best in Class practice questions to determine the areas in which you need to strengthen your skills.

Remember, it’s best to spend some time studying each day instead of cramming for the exam shortly before you take it.

**What test takers wish they would’ve known:**

- Read every answer choice thoroughly, even if you think that the first or second choice is the best answer. Remember that some answer choices may be “almost correct.” You wouldn’t want to accidentally skip the correct answer choice by selecting an “almost correct” answer and moving on without reading the rest of the choices.
- When given passages to read, read over the questions first. By doing this, you’ll be on the watch for relevant information.
- Many test-takers wish they had studied for the ACCUPLACER.

Information and screenshots obtained from the College Board ACCUPLACER website: https://accuplacer.collegeboard.org/sites/default/files/next-generation-test-specifications-manual.pdf

## Reading Placement Test

The Reading Placement Test has about 20 questions.

There are four broad categories:

- Information and Ideas
- Rhetoric
- Synthesis
- Vocabulary

So, let’s talk about Information and Ideas first.

**Information and Ideas**

This category tests your ability to understand what a text states explicitly, as well as what it implies. You will need to be able to identify the overall message, or theme, of a text, as well as draw conclusions.

Let’s look at a concept that will more than likely appear on the test.

**Identifying the Theme of a Text**

Finding a central idea or theme of a text is all about getting to the point. As you read, think about the “big picture” created by the details in the text in order to determine the author’s overall message. Is the author trying to make a general statement about life, or is he/she helping you understand a major concept?

You should be able to determine central ideas and themes, as well as analyze how details in the text support these themes. Remember that a theme is rarely stated outright.

Let’s take a moment to think about Aesop’s fable, *The Crow and the Pitcher*:

*“A thirsty crow found a pitcher with a little water at the bottom. Unfortunately, the pitcher had a narrow neck and the crow couldn’t reach the water. *

*However, the crow had an idea. He began dropping pebbles into the pitcher. With each pebble, the water rose higher. *

*Eventually, the water rose high enough that the crow was able to drink.”*

So, what’s the theme of this fable? Well, Aesop wants us to understand a central message about life. Aesop uses the details in the fable to tell us that *it is better to be determined and resourceful than to accept defeat*. This overall message is the theme of *The Crow and the Pitcher*.

When determining the theme, think beyond the details and what is explicitly stated. How do the details come together to create a larger meaning?

**Rhetoric**

This category tests your ability to analyze both figurative and literal words and phrases and determine how they contribute to the overall mood, tone, and meaning of a text. You’ll also identify and analyze structural formats. On this section, you will need to determine an author’s point of view, purpose, and reasoning.

Here are some concepts you should know.

**Text Structure**

Text structure refers to the way in which the information in a text is organized. Knowing the different types of text structures can help you understand the author’s purpose, the main idea, and the relationship between ideas in a text.

Let’s take a look at some common text structures:

**Description:**In this type of text, the author describes a topic in detail. One example of a descriptive text is an encyclopedia article, regardless of the topic.**Sequence of Events:**The author uses a chronological order to show the sequence in which events happen. An example of this type of text is a passage describing the steps to build a garden fence. Most novels and short stories tend to have a sequence of events structure, as well.**List:**The author uses numerical or chronological order to list items or ideas. For example, most recipes begin with a list of ingredients.**Compare and Contrast:**The author compares and/or contrasts two or more events, objects, or ideas. A comparison shows how two ideas are alike. When an author contrasts ideas, he/she is showing you how the ideas are different. A passage showing the similarities and differences between evergreen and deciduous trees is an example of a text with this structure.**Cause and Effect:**The author introduces one or more causes and then describes the effects of the cause(s). A passage about how reading improves literacy is an example of this text structure.**Problem and Solution:**The author describes a problem or asks a question and then gives possible answers or solutions. An example of this type of writing is a chapter in an article which explains how to treat mosquito bites.

**Point of View**

Point of view refers to the perspective from which a text is narrated. Together, we’ll look at the different types of perspectives a text may be written from, as well as some examples of each type:

**Synthesis**

Synthesis questions will test your ability to understand and describe how similar ideas are expressed by more than one text.

Here are a few examples of tasks you may be asked to perform when answering synthesis questions:

- Compare and contrast the opinions of two authors
- Explain how two passages are related
- Choose a statement that two different authors would be likely to agree/disagree on
- Determine what information is included in one passage, but not in another passage
- Find the main idea of two passages

**Vocabulary**

This category tests your ability to determine the literal and figurative meanings of words and phrases in a text. While you complete vocabulary questions, you will need to use your prior knowledge of terms. You’ll also look at word parts and the context in which terms are used in order to determine their meanings.

The main concept you need to know is how to use context clues.

**Using Context Clues**

Context clues are hints embedded in a text that help readers determine the meaning of a word or phrase. Remember to use context clues when you encounter an unknown word.

Here are a few examples of how to use context clues to find the meaning of unknown words:

**Definition clues:**The author gives the meaning of a term outright. Consider the term “eastern white pine” in the example below.

*“The eastern white pine, a type of tree with long white needles, remains green even in winter.”*

Notice that you need no knowledge of eastern white pines to determine that they are a type of tree.

**Synonym clues:**The author includes a synonym to help the reader understand the meaning of a word.

*“That hotel was luxurious! We were so impressed with how grand and upscale the rooms were.”*

In this example, “upscale” and “grand” can help a reader determine the meaning of “luxurious.”

**Antonym clues:**The author includes an antonym to help the reader understand the meaning of a word.

*“The top of the table was illuminated, but the rest of the room was dark.”*

In this example, “dark” helps the reader determine the meaning of “illuminated.”

And that’s some basic info about the Reading Placement Test.

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

**Directions for questions 1-4**

Read the passage(s) below and answer the questions based on what is stated or implied in the passage(s) and in any introductory material that may be provided.

Passage 1 is by Dorothy Sayers; Passage 2 is adapted from a work by Raymond Chandler.

Passage 1

The detective story does not and cannot attain the loftiest level of literary achievement. Though it deals with the most desperate effects of rage, jealousy, and revenge, it rarely touches the heights and depths of human passion. It presents us with an accomplished fact and looks upon death with a dispassionate eye. It does not show us the inner workings of the murderer’s mind—it must not, for the identity of the criminal is hidden until the end of the book. The most successful writers are those who contrive to keep the story running from beginning to end at the same emotional level, and it is better to err in the direction of too little feeling than too much.

Passage 2

I think what was really gnawing at Dorothy Sayers in her critique of the detective story was the realization that her kind of detective story was an arid formula unable to satisfy its own implications. If the story started to be about real people, they soon had to do unreal things to conform to the artificial pattern required by the plot. When they did unreal things, they ceased to be real themselves. Sayers’ own stories show that she was annoyed by this triteness. Yet she would not give her characters their heads and let them make their own mystery.

## Reading Practice Questions

### Question 1

Which of the following best paraphrases the main idea in Passage 1?

- Detective stories have plots without direction.
- Detective stories may someday be improved upon.
- Detective stories don’t have relatable characters.
- Detective stories are not great literature.

### Question 2

Which of the following statements would the author of Passage 2 most likely agree with?

- Detective writers should hold on tightly to their plots and characters’ actions.
- Dorothy Sayers writes formulaic and predictable detective stories.
- Detective story writers should ensure that the characters are not too deeply developed.
- Dorothy Sayers’ writings have inspired a new type of detective story.

### Question 3

Which of the following best describes the relationship between the two passages?

- Passage 1 focuses on the uniqueness of the genre; Passage 2 focuses on its weaknesses
- Passage 1 discusses the limitations of the genre; Passage 2 discusses the uniqueness of the genre
- Passage 1 highlights the merits of the genre; Passage 2 downplays its faults
- Passage 1 relates the details the particulars of the genre; Passage 2 generalizes the limitations

### Question 4

Which of the following would the author of Passage 1 most likely agree with?

- The characters in detective stories are well-developed and have deep inner worlds
- The detective genre involves a creative and unpredictable writing style
- The plot of a detective story is formulated
- Detective story writers will soon be known as world-class writers

**Directions for questions 5-6**

Read the passage(s) below and answer the questions based on what is stated or implied in the passage(s) and in any introductory material that may be provided.

While most people can name plenty of their favorite artists, ask someone what makes an artist great, and you’ll likely get a different answer from each person you ask. Try to compare the greatness of different artists and you might start an argument. That’s because feeling connected to a work of art is an incredibly personal experience. The same piece of work may affect two people in very different ways, ranging from delight to indifference to disgust. Some works of art end up in the trash, some incite riots, and some are put on the cover of magazines. Still, the art that ends up in the trash could be discovered and treasured years later, while the art on the magazine cover can end up forgotten. No matter what happens to the art, as long as it exists, it always has the potential to inspire others.

### Question 5

The author’s main purpose in this selection is:

- to show that responses to art are subjective and personal.
- to show which qualities make art great.
- to show that people don’t care about art.
- to show that artistic standards don’t change.

### Question 6

According to the information presented in the selection, people disagree on the greatness of art and artists because:

- art has the potential to inspire others.
- the standards of great art haven’t changed since the Renaissance.
- feeling connected to art is a personal experience.
- art is irrelevant in society today.

**Directions for questions 7-8**

Read the passage(s) below and answer the questions based on what is stated or implied in the passage(s) and in any introductory material that may be provided.

1 President Kennedy was not the first to imagine sending a man to the moon. A little more than 100 years earlier, in 1865, science fiction writer Jules Verne also imagined space travel. He put his innovative thoughts in a book called *From the Earth to the Moon*. In it he described a lunar expedition that is so eerily close to the Apollo 11 mission that a reader would think he was predicting the future. He called his spaceship with a crew of three the Columbiad. In his book the spacecraft launches from Florida, and the United States Navy recovers it from the Pacific Ocean. In 1969, Florida was the launch site of Apollo 11. The command module was named Columbia. When the spacecraft returned to Earth, it splashed down in the Pacific, where the Navy recovered it along with its three-astronaut crew. Verne accurately delineated the future when the technology of his own time made his predictions seem highly unlikely to occur. How could he have known that his far-fetched idea was not so far-fetched after all?

2 Like Verne, other science fiction writers have accurately described inventions that are commonplace today. Many of H. G. Wells’s ideas, for example, have become a reality. Considered by many to be one of the best science fiction writers of all time, Wells wrote about lasers, wireless communication, automatic doors, and other gadgets that did not exist at the time of his writings. But today these gadgets are such an integral part of our society that we probably cannot imagine living without them. Wells also describes a journey to the moon on a spaceship made from anti gravity material. We can only speculate that these writers might have inspired those who later turned their fiction into reality.

3 In 2012, a Mars rover, developed by the National Aeronautics and Space Administration (NASA), landed on the planet Mars. No one would have been more excited to hear the news than Ray Bradbury, one of America’s greatest science fiction writers. In 1950 he wrote about travel to Mars in his book *The Martian Chronicles*. The book describes an expedition that lands humans on Mars. The story then tells how the people inhabit the planet and bring their families to live there. Since NASA has successfully landed a rover on Mars, Bradbury’s fantasy may yet become reality. The Mars rover, appropriately called Curiosity, is gathering information that will help NASA plan a manned mission to Mars sometime in the 2030s. Will future families travel to Mars to live there, as Bradbury imagined? If so, the world as we know it today will certainly be different.

### Question 7

What is one detail that illustrates how Jules Verne’s book connects with the real Apollo 11 mission?

- When the spacecraft returned to earth, it landed in the Pacific Ocean
- President Kennedy was in charge of the space launch
- When they landed on Mars, it looked eerily similar to the way Verne had described it
- The mission’s name, Apollo 11, was taken from Verne’s book

### Question 8

How is the information in this selection organized?

- Cause/Effect
- Problem/Solution
- Compare/Contrast
- Spatial

**Directions for questions 9-10**

The following sentences have a blank indicating that something has been left out. Beneath the sentence are four words or phrases. Choose the word or phrase that, when inserted in the sentence, best fits the meaning of the sentence as a whole.

### Question 9

After hearing the testimony of more than ten women, the head detective decided to keep investigating in order to determine if anyone else, in addition to the main suspect, was ____________.

- irreproachable
- culpable
- incognizant
- uninformed

### Question 10

The petulant worker, who was ____________ throwing tantrums and treating guests unfairly, demanded a tip for carrying the couple’s luggage.

- known for
- indebted to
- pleased with
- described as

## Writing Placement Test

The Writing Placement Test has about 25 questions.

There are six broad categories:

- Development
- Organization
- Effective Language Use
- Sentence Structure
- Conventions of Usage
- Conventions of Punctuation

So, let’s talk about Development first.

**Development**

This category tests your ability to revise passages and sentences.

Let’s take a look at a concept that is more than likely to appear on the test.

**Focus**

A text with a clear focus does not include information that strays from the main idea. Sentences that don’t support the main idea serve no real purpose and confuse the reader. On the test, you’ll encounter focus questions which will ask you to add, revise, delete, or retain information in a passage in order to ensure that the text does not stray from its purpose or main idea.

Here are a few ways that you can revise a passage to ensure that it has a clear focus:

- Add a topic sentence if the passage does not include one.
- Delete details that are not relevant.
- Change the wording of a sentence so that its relationship to the main idea is stronger.
- Add sentences that clarify the relationship of details with the main idea.

Let’s take a look at an example passage together:

*The planet Jupiter has a mass which is more than 300 times the mass of the Earth. Jupiter was named after the Roman god of the sky. **In ancient Greece, this god was known as Zeus. Both gods, Zeus and Jupiter, were said to use a lightning bolt as a weapon.** The planet Jupiter is the fifth planet from the Sun, and it is about 150 times farther from the sun than the Earth; therefore, the surface of Jupiter is much colder than the surface of Earth. Most images of Jupiter show the planet as being white, orange, and brown.*

Take a second look at the underlined sentences. What in the world are they doing there? They certainly don’t help you learn about the planet Jupiter, which is the topic of the passage!

The author should delete these two sentences to maintain focus.

**Organization**

This category tests your ability to revise passages as needed so that information and ideas appear in the most logical order. You’ll answer questions about introductions, conclusions, and transitional sentences, phrases, and words.

Here is a concept you should know.

**Sequencing Ideas and Events**

Proficient writers are able to order their ideas logically. In a nonfiction text, writers generally introduce a main idea, give details to support and explain the main idea, and then give the reader a concluding message about the main idea. In both fiction and nonfiction texts, writers often need to explain events in the order in which they occur.

When reading a passage on the Writing Placement Test, ask yourself what the purpose of each sentence is. Is the sentence clarifying the previous sentence? Is it introducing a new idea? Is it showing the connection between two ideas? Determining the purpose of each sentence allows you to decide whether or not it is correctly placed in a passage.

When completing the test, you may be asked whether or not a sentence in a passage should be moved to a different place in the passage. Remember that you can take your time on this test, so try the sentence out in each suggested place in the passage. Ask yourself where the sentence makes the most sense. Usually, your “gut feeling” is correct.

**Effective Language Use**

This category tests your ability to revise a text by replacing existing words with more precise and appropriate words. You’ll also show that you can identify words that should be deleted, because they are pointless or redundant. Furthermore, you will show that you are able to create and recognize a variety of sentence structures.

Take a look at this concept.

**Sentence Structures**

In order to perform well on this section, you will need to be able to identify and create sentences with a variety of structures. Let’s take a look at some different types of sentence structures, as well as some examples of each:

A **simple sentence** has only one clause. A clause is a phrase which contains a subject and a verb. The clause in a simple sentence is called an independent clause, because it is able to stand alone. Here are examples of simple sentences:

*He is cooking chicken for dinner tonight.*

*This soup contains fresh vegetables, a variety of savory herbs, and oven-roasted chicken.*

A **compound sentence** contains two or more independent clauses, joined by a conjunction. Let’s look at some examples of compound sentences. Notice that the conjunctions between clauses are underlined.

*Rob wants to visit New York City,***so**he will buy a plane ticket.

*The boy did not want to tattle on his friend,***nor**did he want to feel guilty about staying silent.

*Alice definitely did not want to go to the movie theater, but she did not feel like going to a restaurant either,***so**she ended up staying home.

**Complex sentences** have at least one independent clause, as well as at least one dependent clause. Unlike independent clauses, dependent clauses cannot stand alone. A dependent clause is missing either a subject or a verb, or it contains a subject and a verb, but it is not a complete thought. In the following examples of complex sentences, the dependent clauses are underlined:

**Despite Ian’s doubts**, he found that his new professor was very agreeable.

**When I went to visit my cousin, who lives in Greensboro**, I met some of her friends.

**Compound-complex sentences** have at least two independent clauses, as well as at least one dependent clause. In the following examples, the dependent clauses are underlined:

*I want to watch television,***though considering the state of the house**, I need to finish cleaning the bedroom first.*After Sarah’s dalmation got sick**, she started choosing pet food more carefully, and her dog is perfectly healthy now.*

**Sentence Structure**

This category tests your ability to recognize and appropriately use verb tenses, pronouns, and punctuation. You will also be tested on your knowledge of sentence structures, subordinate and coordinate clauses, and modifiers.

Take a look at this concept that may come up on the test.

**Modifiers**

A modifier is a word, phrase, or clause that describes another word. Usually, modifiers are adjectives, like “friendly” or “personable.” Sometimes, a modifier is included in the wrong part of a sentence. Consider this example:

“The **live** bowl of fish looks great in her office.”

Since the fish are alive and the bowl is not, “live” is a misplaced modifier. You could revise the sentence by moving the modifier:

“The bowl of **live** fish looks great in her office.”

A dangling modifier is a modifier that does not have a clear subject. The word that a dangling modifier is meant to describe is missing. Consider this example:

“**Staring** over the fields, the storm approached.”

“Staring” is a modifier, but who or what is staring? There are multiple ways to revise this sentence, and here is one example:

“**Uncle George was staring** over the fields as the storm approached.”

**Conventions of Usage**

This category tests your ability to correctly use words that are commonly misused, revise expressions, and correct errors in the agreement between subjects and verbs and pronouns and antecedents.

Let’s look at a concept that is more than likely to pop up on the test.

**Pronoun-Antecedent Agreement**

**Pronouns**, such as “he,” are used to replace more specific **nouns** like “Barkley.” If a more specific noun, like “Barkley,” occurs first in the text, then that noun is an **antecedent**. Consider this information as you read the following sentence:

*“If **Barkley** could speak, **he** would probably ask you to get **his** leash and take him for a walk in the dog park.” *

In the example, “Barkley” is the antecedent and “he” and “his” are pronouns. Pronouns should agree with their antecedents in terms of number and gender.

**Conventions of Punctuation**

This category tests your ability to correctly use punctuation to end sentences, write lists, show possession, and separate ideas.

Here is a concept to know.

**Semicolons**

We tend to use semicolons (;) less often than many other types of punctuation marks; therefore, many people are confused about how to use them.

Think back to when you read about independent clauses, which are able to stand on their own. We often use conjunctions like “and” and “for” to join independent clauses. Sometimes, however, it may be appropriate to join two independent clauses using a semicolon. A semicolon divides two independent clauses, but it tells you that the information in those clauses is related.

Here are a few examples of appropriate semicolon use to separate independent clauses:

*I know you don’t like going to the gym; nevertheless, it is very good for **you.*

*We should plan to see each other soon; we haven’t spent any time together lately.*

Another appropriate time to use semicolons is when separating items in a list that already contains commas. Here’s an example:

*Andrea visited Lima, Peru; Santiago, Chile; and Caracas, Venezuela.*

And that’s some basic info about the Writing Placement Test.

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

## Writing Practice Questions

**Directions for questions 1-3**

Read the following early draft of an essay and then choose the best answer to the question or the best completion of the statement.

(1) Sports are a wonderful means for mankind to exercise one of its most basic principles: competition with our fellow man. (2) Surrounding all types of sports is the concept of sportsmanship – the respect and ethical behavior shown to all participants of a contest. (3) The spirit of the game, in many cases, is more important than the outcome of the match; a true competitor understands this. (4) This is why many of our most beloved athletes are not always the most talented performers—it is the players who play with the purest motive, for the sake of the team, and with respect for all opponents, who gain the respect and admiration of the fans.

(5) There are greater lessons to be learned from sports than being well liked by fans. Sports, and by extension, the athletes who play them, extend beyond cultural differences; surely styles of play can vary between countries and regions, but in general, sports are played the same everywhere. (5) Similarly, fans of a sport are able to appreciate incredible athletic feats or displays of true sportsmanship regardless of the player. (6) Simply put, in a day and age when settling cultural differences is of utmost importance, sports are a reasonably viable way to bring the world closer together.

(7) Lastly, international events such as the Olympic Games or World Cup are perfect opportunities to show the world that international cooperation and peace are possible. (8) Sports can and should be used as instruments of change in an uncertain world. (9) They can also be proponents of peace.

(10) The Olympic Creed says it best: “The most important thing in the Olympic Games is not to win but to take part, just as the most important thing in life is not the triumph, but the struggle; the essential thing is not to have conquered, but to have fought well.”

### Question 1

Which transition would best connect the mention of sportsmanship in paragraph one to the first sentence of paragraph two?

- In other words
- Therefore
- For example
- However

### Question 2

Select the option which is the best version of the sentence reproduced below.

*Surrounding all types of sports is the concept of sportsmanship – the respect and ethical behavior shown to all participants of a contest.*

- (As it is now)
- Surrounding all types of sports is the concept of sportsmanship – the respect and ethical behavior shown to all participants of a contest – is sportsmanship.
- The concept of sportsmanship the respect and ethical behavior shown to all participants of a contest, surrounds all sports.
- Sportsmanship surrounds all sports and it is defined as the respect and ethical behavior shown to all participants of a contest.

### Question 3

Which of the following is the most effective way to revise and combine the following sentences from paragraph three? “Sports can and should be used as instruments of change in an uncertain world. They can also be proponents of peace.”

- (As it is now)
- Sports can and should be used as instruments of change in an uncertain world, and they can also be proponents of peace.
- Sports can and should be used as instruments of change in an uncertain world; they can also be proponents of peace.
- Sports can and should be used as instruments of change and proponents of peace in an uncertain world.

**Directions for questions 4-10**

Read the following early draft of an essay and then choose the best answer to the question or the best completion of the statement.

(1) It is blue at one time and green at another, even from the same point of view. (2) Lying between the earth and the heavens and it partakes of the color of both. (3) When viewed from the top of a hill it reflects the color of the sky but near at hand it is of a yellowish tint next the shore where you can see the sand, then a light green, which gradually deepens to a uniform dark green in the body of the pond.

(4) Like the rest of our waters, when much agitated, in clear weather, so that the surface of the waves may reflect the sky at the right angle, or because there is more light mixed with it, it appears at a little distance of a darker blue than the sky itself and at such a time, being on its surface, and looking with divided vision when I see the reflection, I have discerned a matchless and indescribable light blue, such as watered or changeable silks and sword blades suggest, more cerulean than the sky itself, alternating with the original dark green on the opposite sides of the waves, which last appeared but muddy in comparison.

(5) It is a vitreous greenish blue, as I remember it, like those patches of the winter sky seen through cloud vistas in the west before sundown. (6) How large a body of Walden water would be able to reflect a green tint I have never proved. (7) The water of the local river is sometimes even black or a very dark brown to one looking directly down on it. (8) The water of Walden Pond’s is of crystalline purity.

### Question 4

Which sentence from the passage includes a pronoun with no clear antecedent?

- Sentence 1
- Sentence 2
- Sentence 3
- Sentence 4

### Question 5

Which of the following sentences from a passage is a run-on sentence and should be broken into multiple sentences?

- Sentence 1
- Sentence 2
- Sentence 3
- Sentence 4

### Question 6

Which is the best version of the underlined portion of sentence 2 (reproduced below)?

*Lying between** the earth and the heavens and it partakes of the color of both. *

- (As it is now)
- The pond lies
- It is lying between
- Between both

### Question 7

Which is the best version of the underlined portion of sentence 3 (reproduced below)?

*When viewed from the top of a hill it reflects the color** of the sky but near at hand it is of a yellowish tint next the shore where you can see the sand, then a light green, which gradually deepens to a uniform dark green in the body of the pond. *

- (As it is)
- When viewed, from the top of a hill, it reflects the color
- When viewed from the top of a hill, it reflects the color
- When viewed from the top of a hill it reflected the color

### Question 8

Sentence 3 is reproduced below.

*When viewed from the top of a hill it reflects the color of the sky but near at hand it is of a yellowish tint next the shore where you can see the sand, then a light green, which gradually deepens to a uniform dark green in the body of the pond. *

The writer is considering adding the following text sentence to the end of the first paragraph after this sentence.

*Any swimmer who enters the pond appears to have his limbs magnified and distorted by the waves; this produces a monstrous effect, making fit studies for a Michael Angelo.*

- Yes, because it establishes that the author is not the only person to have visited the pond.
- Yes, because it helps to connect the ideas in the first paragraph with the ideas in the second paragraph.
- No, because the language is much more descriptive than the rest of the language in the first paragraph.
- No, because it strays from the main focus of the first paragraph.

### Question 9

Which is the best version of the underlined portion of sentence 6 (reproduced below)?

*How large a body of Walden water** would be able to reflect a green tint I have never proved.*

- (As it is)
- How large of a body of Walden water
- How a large body of water, like Walden,
- How large a body of water as Walden

### Question 10

Which sentence blurs the focus of the last paragraph and should be deleted?

- Sentence 5
- Sentence 6
- Sentence 7
- Sentence 8

## Math: Arithmetic Placement Test

The Math: Arithmetic Placement Test has about 20 questions.

There are five broad categories:

- Whole Number Operations
- Fraction Operations
- Decimal Operations
- Percent
- Number Comparisons and Equivalents

So, let’s talk about Whole Number Operations first.

**Whole Number Operations**

This category tests your ability to add, subtract, multiply, and divide whole numbers. You will need to use the order of operations. You’ll also need to know how to estimate values and round answer choices when necessary.

Let’s look at a concept.

**Order of Operations**

“PEMDAS” is the acronym which is often used to remember the order of operations. This acronym stands for Parentheses, Exponents, Multiplication/Division, and Addition/Subtraction. The mnemonic phrase, “**P**lease **E**xcuse **M**y **D**ear **A**unt **S**ally” can help you remember “PEMDAS” while taking the test.

Here’s a simple example problem:

*7 + 5 – (4 × 2)*

Solve what’s in the parentheses first:

*7 + 5 – 8*

Now, we add 7 and 5 to get 12:

*12 – 8*

Finally, we subtract 8 from 12 to get our answer:

*4*

**Fraction Operations**

This category tests your ability to solve problems with fractions and mixed numbers. On this section, you will need to apply your knowledge of the order of operations, estimation, and rounding. You will be asked to add, subtract, multiply, and divide fractions and mixed numbers.

Take a look at this concept.

**Dividing Fractions by Fractions**

When presented with a question that asks you to divide one fraction by another fraction, keep in mind that dividing fractions is just multiplying with reciprocals (inverses). Once your multiplication problem is set up, you will multiply the numerators, or values on top, by one another. Then you will multiply the denominators, or the values on bottom, by one another.

What might this look like on the test? Here’s an example:

8/6 *÷ *3/2

First, set up your multiplication problem by flipping the second fraction (finding the reciprocal/inverse):

8/6 *× *2/3

Multiply across to find your answer:

16/18 *or *8/9

You have your answer!

**Decimal Operations**

This category tests your ability to use the order of operations to add, subtract, multiply, divide, and round numbers that contain decimals. You may also need to convert decimals into fractions.

Let’s look at a concept that will more than likely appear on the test.

**Converting Decimals to Fractions**

On the test, it is likely that you will need to convert decimals to fractions. You may need to do this to add, subtract, multiply, or divide decimals and fractions within the same problem.

Converting decimals to fractions may sound complicated if you haven’t done it before, but we can do it in just three easy steps. Let’s practice by converting .55 to a fraction.

**Step 1: **Write the decimal as a numerator over the number 1:

.55/1

**Step 2: **Multiply both the numerator and the denominator by 10 for every digit to the right of the decimal. Since .55 has **two** digits to the right of the decimal, we will multiply both the numerator and the denominator by 10 **twice**:

*(.55 x 10 x 10 = 55) / (**1 x 10 x 10 = 100)*

*or*

*55 /**100*

**Step 3:** Now that we have converted .55 to a fraction, we need to simplify it. Look for the largest number that we can divide both 55 and 100 by to get two whole numbers.

In this case, 5 is the largest number that we can divide both 55 and 100 by to get whole numbers. When we divide both the numerator and denominator by 5, we get our final and simplified answer:

*11 / **20*

**Percent**

This category tests your ability to work with percentages in real-life contexts, such as calculating prices. You’ll need to be able to determine certain percentages of given numbers. You will also be asked questions that require you to calculate the percentage by which numbers decrease or increase.

Here is a concept you should know.

**Calculating Percent Increase/Decrease**

Whether calculating percent increase or decrease, you will be using the same formula. After all, you are really just calculating the percentage of change, regardless of whether there is an increase or decrease in value.

Here’s what the formula looks like:

So, let’s put the formula to work with an example:

*Last semester, 75 freshmen enrolled in a drawing course. This semester, 81 students have enrolled in the same drawing course. What is the percentage increase in drawing course enrollment?*

So, we will subtract the old value from the new value. Note that you do **not** need to worry about whether the answer is positive or negative. Just consider the **absolute value** of the answer:

*81 – 75 = 6*

Next, we will divide 6 by the old value:

*6 ÷ 75 = .08*

The final step is to multiply that answer by 100.

*.08 x 100 = 8*

Therefore, there was an 8% increase in student enrollment in the drawing course.

Remember that you will use the same formula to calculate percent decrease and percent increase. Isn’t it such a relief that you don’t have to memorize another formula?

**Number Comparisons and Equivalents**

This section will test your ability to compare and order numbers. You’ll need to remember how to use equality and inequality symbols, as well as how to correctly place numbers on a number line.

Let’s look at some concepts.

**Ordering Numbers**

The easiest way to order numbers is to convert any numbers containing fractions to decimals; then plot the numbers on a number line. Let’s look at an example together:

Let’s say you are given the following numbers:

*-½, .5, 1¼ , 2.75*

Let’s convert them each to decimals:

*-.5, .5, 1.25, 2.75*

Numbers are plotted on a number line by using a dot or circle for each number. Take a look at the following number line. Notice that the place of each of our numbers (*-.5, .5, 1.25, 2.75) *is indicated as a point on the number line.

**Comparing Numbers**

Whether you are comparing whole numbers, fractions, decimals, or mixed numbers, you will use the same comparison symbols:

**Less than: <**

*Examples: *

2 < 5

.10 < .20

5 ½ < 10 ¾

½ < ¾

**Greater than: >**

*Examples: *

*21 > 5*

.10 > .01

51 ½ > 2¾

½ > *⅓*

**Equal to: =**

*Examples: *

*2 = 2*

.5 = ½

3 ¾ = 3.75

*⅓ = ⅓ *

It’s easy to remember the greater than and less than symbols, because the larger side of each symbol always points to the larger number!

And that’s some basic info about the Math: Arithmetic Placement Test.

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

**Directions**

Choose the best answer. Use paper if necessary.

## Mathematics Arithmetic Practice Questions

### Question 1

The varsity basketball team has 3 freshmen, 5 sophomores, 3 juniors, and 4 seniors. Approximately what percentage of the basketball team is comprised of sophomores?

- 30%
- 33%
- 25%
- 20%

### Question 2

Which of the following is equivalent to 6 + (5 – 2)² × 3?

- 45
- 36
- 24
- 33

### Question 3

Which of the following fractions is equal to 0.08?

- 1/8
- 1/80
- 8/10
- 8/100

### Question 4

A horse trotted 7 2/3 miles in 3/4 of an hour. What was the horse’s average trotting speed, in miles per hour?

- 11 5/6
- 9 1/2
- 8 5/7
- 10 2/9

### Question 5

Cierra purchased a car for $22,000. She is now trying to sell the car for $14,000. By approximately what percent did Cierra reduce the price of her car?

- 57%
- 8%
- 80%
- 36%

### Question 6

4/7, 5/11, 3/5, 7/9

Which of the fractions above is the greatest?

- 4/7
- 7/9
- 3/5
- 5/11

### Question 7

Which of the following points on a number line is the greatest distance from 0.5?

- 1.5
- 0
- -1.5
- 1

### Question 8

What is 2,596 + 853?

- 3,449
- 3,349
- 3,309
- 4,449

### Question 9

What is the value of 4.83 + 0.006 + 0.135?

- 6.24
- 5.24
- 4.971
- 4.961

### Question 10

What is 0.5436 rounded to the nearest hundredth?

- 0.55
- 0.544
- 0.54
- 0.543

## Math: Quantitative Reasoning, Algebra, and Statistics Placement Test

The Math: Arithmetic Quantitative Reasoning, Algebra, and Statistics Placement Test has about 20 questions.

There are ten broad categories:

- Rational Numbers
- Ratio and Proportional Relationships
- Exponents
- Algebraic Expressions
- Linear Equations
- Linear Applications and Graphs
- Probability and Sets
- Descriptive Statistics
- Geometry Concepts for Pre-Algebra
- Geometry Concepts for Algebra 1

So, let’s talk about a few of the more challenging categories.

**Algebraic Expressions**

This category tests your ability to create and solve algebraic expressions and identify equivalent expressions. On this portion of the test, you will also need to combine like terms in order to simplify equations and determine solutions.

Take a look at this concept.

**Identifying Equivalent Expressions**

Equivalent expressions have the same solution, but they may be written differently. To solve for equivalent expressions, you’ll need to combine like terms and use the distributive property.

Here’s an example of two equivalent expressions:

*2x**⁴** + 2x* and *2(x**⁴ **+ x). *

The second expression has the same value as the first. If you distribute the 2 in the second expression, it will look like the first one.

In order to find out if two expressions are equivalent, just remember to simplify each expression as much as possible by combining like terms.

**Linear Equations**

This category tests your ability to solve and simplify linear equations and inequalities. You may be asked to solve systems of linear equations.

Let’s look at a concept.

**Solving Linear Equations**

A linear equation has simple variables, such as x and y. We will explore graphing linear equations later on. For now, let’s look at an example and solve it:

*3x + 15 = x + 25*

Now remember that the equals sign tells you that the values on each side are equivalent. If we are going to solve for x, we need to combine like terms. Let’s start by moving 15 to the other side of the equation.

When we move 15 to the other side, we need to use its inverse (-15).

*3x + 15 = x + 25*

*-15 -15*

*3x = x + 10*

Next, we need to get all of our x variables on one side of the equation. Remember that *“x”* is actually *“1x.”* Therefore, we need to subtract *1x* from the left side of the equation:

*3x = 1x + 10*

*-1x -1x*

*2x = 10*

Our last step is to get a single* x* variable alone. We can do this by dividing each side of the equation by 2:

*x = 5*

**Linear Applications and Graphs**

This category tests your ability to graph and describe linear equations. You will work with systems of equations, as well as linear inequalities.

Let’s look at a concept that will more than likely pop up on the test.

**Graphing Linear Equations**

To graph a linear equation, we must determine the value of each variable. Let’s graph *x + 2y = 7* together. In order to graph this equation, we will set each of the variables to 0.

First, we will set* x* as 0. Our equation becomes:

0 + 2y = 7

In order to solve for y, we just need to divide each side by 2:

y = 3.5

Now that we have determined the value of y, we will determine the value of x. We can do this by setting y to 0 in the original equation:

*x + 2(0) = 7*

*or *

*x + 0 = 7*

*therefore,*

*x = 7*

We have now determined the value of both *x *and *y*: ** x = 7 **and

*y = 3.5.*Now, let’s think about graphing. Remember that a coordinate on a graph is always shown as (*x, y*).

We know that two of our points will be (0,3.5) and (7,0). Using a graph, we can mark each of these two points and draw a line between them:

**Probability and Sets**

This category tests your ability to calculate probability, both when events are related and when they are independent from one another. You’ll show that you can answer questions about simple, compound, and conditional probability.

Here is a concept you should know.

**Conditional Probability**

Unlike compound probability, conditional probability refers to the likelihood that two related events will occur. When working with conditional probability, events are dependent, as opposed to independent. Let’s look at an example:

*Several children go to a local fair and play a game. While playing the game, each child has two chances to throw a ball at a target. *

*Out of all the children, 35% hit the target on the first try, and 15% hit the target on both tries *(these are the given conditions – that’s where the “conditional” part comes in).

*Given these conditions, what is the probability that a child who hits the target on the first try also hits the target on the second try?*

In order to answer this question correctly, we need to use an equation which can be used to determine conditional probability:

*P(A|B) =* *P(AandB)/**P(B)*

This equation tells us that the probability of B occurring *if *A has occurred is the probability of both events occurring divided by the probability of B occurring.

Each time a child attempts to hit the target is an event. 15% of the children hit the target twice (event A and event B). 35% of the children hit the target on the first try (event A). Therefore, our formula will look like this:

*P(A|B) = **15/**35*

When we divide 15 by 35, we get .4285.

Now, we need to show that decimal as a percentage: 42.85%. Our answer shows that any child who hits the target on the first try has a 42.85% chance of hitting the target on the second try.

**Descriptive Statistics**

This category tests your ability to work with statistics that describe various data sets. You will use measures of center (mean, median, and mode) to describe the data that is presented to you. You should also be able to interpret data presented in a variety of graphical displays.

Here is a concept you should know.

**Measures of Center**

On the test, you’ll answer questions about measures of center, such as mean, median, and mode. Take a look at the following set of numbers:

*2, 2.5, 3, 3, 3.5, 7*

The **mean** is the average. Add all of the numbers in the set together, and then divide by the amount of numbers in the set (in this case 6). The mean of this list is 3.5.

The **mode** is the number that occurs most in the set. In this case, the only number that appears more than once is 3. Therefore, 3 is the mode.

The **median** is the number that appears in the middle of the list when the numbers are ordered from least to greatest. What do you do if you have an even amount of numbers and no “middle” number? You add the **two** numbers in the center and divide them by two. Since *3 + 3 = 6* and *6/2 = 3*, 3 is the median of this particular set.

**Geometry Concepts for Pre-Algebra**

This category tests your ability to understand and determine the radius, diameter, perimeter, and circumference of circles. You will also use formulas to determine the volume of prisms.

Let’s look at a concept that may appear on the test.

**Circumference of a Circle**

To calculate the circumference of a circle, you can use the formula *C = 2πr.* The “r” represents the radius, a segment that connects the center of the circle to the edge of the circle. Alternately, you can use the formula *C = πd. *The “d” in that formula represents diameter, which is twice the radius.

How might this appear on the test?

*You have a circular flower bed with a radius of 4 feet. *

*What is the circumference of the flower bed?*

A. 25.13 ft.

B. 50.27 ft.

C. 30.17 ft.

D. 21.22 ft.

The answer to this question is A. In order to find this answer, we plug the value of the radius into the formula *C = 2πr. *Our equation is *C = 2Π**4.*

**Geometry Concepts for Algebra 1**

This category tests your ability to create expressions for area, perimeter, and volume problems. You’ll also use the distance formula and the Pythagorean Theorem to solve problems.

Take a look at this concept.

**Pythagorean Theorem**

You will use the Pythagorean equation (*a² + b² = c²*) to solve problems regarding right triangles.

According to this theorem, side *c* is the side of the triangle opposite the right angle; its name is the *hypotenuse.* Remember that a right angle is exactly 90°. Side* a *and side* b *are the other two sides (the legs) of the right triangle.

Here’s an example of what a related question might look like on the test:

*Jane cuts a square piece of paper into 2 right triangles. She discards one triangle of paper and is left with the other. *

*Side a of the triangle is 4 inches, and side b is 5 inches. If side c is the hypotenuse, what is the length of side c?*

To solve this problem, just plug the numbers into the Pythagorean equation:

*(4² + 5² = c²)*

Apply the exponents:

*(16 + 25 = c²)*

Add the values on the left:

*(41 = c²)*

And find the square root of each side:

*(6.4 = c)*

So, the hypotenuse of Jane’s triangle must be 6.4 inches!

And that’s some basic info about the Math: Quantitative Reasoning, Algebra, and Statistics Placement Test Test.

**Directions**

Choose the best answer. Use paper if necessary.

## Mathematics Quantitative Practice Questions

### Question 1

Which of the following is the equation of a line that passes through (2, -5) and (-3, 10)?

- y = -3x + 1
- y – 3x = 1
- 3x + 2y = 1
- y + 1 = -3x

### Question 2

If 1/6x – 4 = 1, then x =

- 2
- 10
- 30
- -18

### Question 3

Which of the following is equivalent to the expression (4ab)(-5ab)?

- -20a²b²
- -ab
- –a²b²
- -20ab

### Question 4

Which of the following correctly combines like terms for the following expression?

3x + 2(3x – 2) + 5 – 2(2x + 1)

- 5x – 1
- 15x – 1
- 4x + 6
- 13x + 11

### Question 5

A quilter is preparing to make a quilt for a baby’s crib that is 30 inches by 50 inches. The design of the quilt calls for a diagonal stripe of ribbon from one corner of the quilt to another corner as shown in the image above. What is the approximate length of that diagonal stripe of ribbon in the finished quilt?

- 80 inches
- 58 inches
- 40 inches
- 45 inches

### Question 6

Which of the following expressions (in square units) is the most reasonable estimate for the area of the circle in the graph shown above?

- 41
- 128
- 23
- 27

### Question 7

According to the table above, approximately how many stones are in one US ton?

- 224
- 9
- 8
- 143

### Question 8

Marco completed a 130-page book in 100 minutes. The graph above shows the total number of pages he had read in 20-minute increments. Use the graph to determine how many pages Marco read during his last 40 minutes of reading.

- 42 pages
- 62 pages
- 13 pages
- 68 pages

### Question 9

A change purse contains 4 pennies, 3 nickels, 2 dimes, and the rest of the coins are quarters. If a person has a 1/3 probability of selecting a penny when randomly selecting a coin from the change purse, how many quarters are there?

- 4
- 3
- 12
- 2

### Question 10

The dot plot above shows the shoe size of the 25 students in Ms. Redmond’s kindergarten class. What is the median of these shoe sizes?

- 9
- 11
- 10.5
- 13

## Math: Advanced Algebra and Functions Placement Test

The Math: Advanced Algebra Functions Placement Test has about 20 questions.

There are eleven broad categories:

- Linear Equations
- Linear Applications and Graphs
- Factoring
- Quadratics
- Functions
- Radical and Rational Equations
- Polynomial Equations
- Exponential and Logarithmic Equations
- Geometry Concepts for Algebra 1
- Geometry Concepts for Algebra 2
- Trigonometry

So, let’s talk about some of the more challenging categories.

**Factoring**

This category tests your ability to recognize quadratic equations and cubic polynomials, convert those equations to standard form, and factor them out.

Let’s take a look at a concept that is likely to appear on the test.

**Factoring Quadratic Equations**

The standard form of a quadratic equation is *a**²** + bx + c = 0*. In quadratic equations, *a, b, *and* c* are known variables. Keep in mind that *a *is never equal to 0 in a quadratic equation.

Let’s look at an example of a quadratic equation and practice factoring it:

*x**² **+ 6 = 0 -5x*

This is a quadratic equation in disguise. Once we move the x value to the other side, we can see the equation in standard form:

*x**² **+ 5x + 6** = 0 *

In this equation, *a = 1, b = 5, *and* c = 6*. The first step is to find out which factors (numbers multiplied by each other) will give us the *c *value (6 in this case). We can multiply 6 and 1 or 2 and 3 to get 6.

Next, we have to decide which set of factors will give us the* b *value (5) when added together:

*6 + 1 = 7*

*3 + 2 = 5*

Now we know that 3 and 2 must be the factors that we want to work with, because they add up to 5. In order to factor the quadratic equation, we need to plug the factors (3 and 2) into this equation:

*(x + )(x + )*

Therefore, when the equation is factored, we get:

*(x + 3)(x + 2)*

*or*

*(x + 2)(x + 3)*

Notice that the factors may be plugged into the equation in any order.

**Quadratics**

This category tests your ability to create, simplify, and solve quadratic equations and systems of quadratic equations. Remember that a system of equations refers to two or more equations which work together.

Take a look at this concept.

**Simplifying Quadratic Equations**

As we’ve learned, the standard form of a quadratic equation is *a² **+ bx + c = 0*, but sometimes these equations appear in disguise. Let’s look at an example of a quadratic equation that requires multiple steps to simplify into standard form:

4/32 *[(184t**²** – 168t**²**) + (-18t × 4)] + 2 = -13t +7*

First, let’s deal with *4/**32*. We can easily simplify the fraction to *1/**8*:

*1/**8 **[(184t**²** – 168t**²**) + (-18t x 4)] + 2 = -13t +7*

Next, let’s complete the simple arithmetic in the brackets:

*1/**8 **(16t**²** – 72t) + 2 = -13t +7*

Now, let’s multiply the variables in parentheses by *1/**8*:

*2t**²** – 9t + 2 = -13t + 7*

This is starting to look much more like a quadratic equation now, but we still need to combine like terms. Let’s move 13t to the other side of the equation via addition:

*2t**² **+ 4t + 2 = 7*

Next, we need to subtract 7 from each side in order for our equation to equal 0:

*2t**²** + 4t – 5 = 0*

Finally, we have our simplified quadratic equation. We know that *a = 2, b = 4, *and* c = −5.*

**Functions**

This category tests your ability to evaluate and graph linear and quadratic functions. You should be able to determine the function of a variable and use that information to create and identify points on a graph which correspond to that function.

Here is a concept you should know.

**Graphing Functions**

Let’s practice graphing a simple quadratic function together:

*f(x) = x**²*

First, let’s list some points for the x and y values. The function of x will be our y value. We know that the y value of any point will just be the x value squared.

The table that we have created gives us the points for our graph. We can see that one point on the graph is (-2, 4), another point is (-1, 1), and so on. Let’s graph the points:

You just graphed a function! Wasn’t that easy?

**Radical and Rational Equations**

This category tests your ability to work with equations that contain radicals (such as squares, cubes, fourth roots, etc.). You will also work with equations that contain fractions with variables in their denominators. You should be able to create, simplify, and graph these equations.

Here is a concept you need to know.

**Simplifying Radical Equations**

Let’s practice working with radical equations by simplifying:

3√405t⁶y⁴

First, we need to break down 405 into its prime factors:

*405 = 3 •** 3 • ** 3 •** 3 • ** 5*

Now, let’s factor out *t⁶ *and* y⁴*:

*t⁶** = t • ** t • ** t • ** t • ** t • ** t*

*y⁴** = y •** y • ** y • ** y*

Next, we need to look at the index of the radical. In this case, our index is 3, because we are working with a cube root. This means that we need to identify any factors that appear in a group of three. These factors are underlined for you in the example below:

3√405 *= **3 •** 3 • *** 3** • * 3 • ** 5*

3√t⁶ *=** t• ** t• ** t•* *t• ** t• ** t*

3√y⁴ * = **y •** y • *** y **•* y*

Now, all of the underlined factors need to be moved outside of the radical. We will move *y* and 3 out once each (because they each have one group of 3). Because there are two groups of *t*, we will move that factor out twice:

3 *• *t *• *t *• *y (3√3 *• *5 *• *y)

Notice that any variables that are not underlined remain inside the radical. The last step is to simplify the expression outside of the radical, as well as the expression inside of the radical. When we simplify by multiplication, our final expression is:

3t²y³√15y

**Polynomial Equations**

This category tests your ability to solve equations with multiple variables, as well as polynomial equations. You will also be expected to graph polynomial functions.

Let’s look at an important concept.

**Solving Polynomial Equations**

Here’s an example of a polynomial equation. We’ll solve it step by step:

*5y**²** + 3(5y) = 0*

First, we need to complete the arithmetic inside the parentheses:

*5y**²** + 15y = 0*

Next, we will factor out 5*y,* because both variables on the left side of the equation contain 5*y*:

*5y (y + 3)= 0*

Now, we will use the zero principle. Set each factor equal to 0:

* 5y = 0*

*y + 3 = 0*

We can solve 5y = 0 by dividing each side by 5 to get *y = 0*. The equation *y + 3 = 0 *can be solved by subtracting 3 from each side:* y = -3.*

Therefore, *y = 0 *or* y = -3*.

**Exponential and Logarithmic Equations**

This category tests your ability to solve and graph equations that contain exponents or logs. Keep in mind that logs are the inverse of exponents. You will also demonstrate that you know how to graph equations that contain logs.

Let’s look at a concept.

**²Solving Logarithmic Equations**

Take a look at the following equation:

*log*_{b}(x*²** – 30) = log*_{b}*(x) *

_{b}(x

_{b}

Since the logs both have the same base, we can set both sides equal to each other:

*x**²** – 30 = x*

Let’s set the equation to 0 by subtracting the x from both sides. Then, we can factor:

*x² – x – 30 = 30*

*(x – 6)(x + 5) = 0*

*x = *6 or -5

Now, we have to decide if x is equal to 6 or -5. Since a log cannot contain a negative number, x is equal to 6.

**Geometry Concepts for Algebra 1**

This category tests your ability to interpret and measure geometrical shapes and their dilations, rotations, translations, and reflections.

Here is an important concept.

**Dilations**

To dilate something means to make it larger. A dilation of a geometric shape is the same geometric shape, only its size has increased. Consider figure F on the graph below:

Figure* L* is a dilation of figure *F*. In order to get figure* L*, we multiply the distance between the points in figure *F* by 2.

*Side a = side w*

*Side b = side x*

*Side c = side y*

*Side d = side z*

Figures* H *and* K* are rotations of *F*. Figure *P* is larger than* F*, but it is not a dilation, because its proportions have been distorted.

**Geometry Concepts for Algebra 2**

This category tests your ability to work with equations and theorems in order to evaluate lines and angles. You’ll also determine the measures of shapes and look for instances of congruency.

Let’s take a look at a concept that may appear on the test.

**Intersecting Line Theorems**

Let’s explore theorems related to parallel lines and transversals. Transversals are lines that intersect parallel lines. Take a look at the example of a transversal and two parallel lines below:

In order to understand some theorems, let’s identify the angles in the example:

**Alternate interior angle pairs:** (3 and 6) (4 and 5). These angle pairs are on opposite sides of the transversal, and they are between the parallel lines.

**Alternate exterior angles:** (1 and 8) (2 and 7). These angle pairs are on opposite sides of the transversal, and they are outside of the parallel lines.

**Corresponding angles:** (1 and 5) (2 and 6) (3 and 7) (4 and 8). These angle pairs are corresponding, because each angle is in the same position in its group of four angles.

**Same-side interior angles:** (3 and 5) (4 and 6). These angle pairs are on the same side of the transversal, and they are between the parallel lines.

**Same-side exterior angles:** (1 and 7) (2 and 8). These angle pairs are on the same side of the transversal, and they are outside of the parallel lines.

Now that you can identify the angles, you can apply them to the following criteria in order to determine whether lines are parallel:

If parallel lines are intersected by a transversal,

1. corresponding angles are congruent

2. alternate interior angles are congruent

3. alternate exterior angles are congruent

4. same-side interior angles are supplementary

5. same-side exterior angles are supplementary

Let’s use Rule 4 as an example. If angles 3 and 5 add up to 180°, then we know that we are working with a set of parallel lines. If they add up to a different number, the lines aren’t parallel.

**Trigonometry**

This category tests your ability to evaluate and graph trigonometric equations, functions, and relationships. In order to do well on this portion of the test, you will need to be able to determine the length of arcs and use the law of sines and the law of cosines.

Take a look at this concept.

**Determining Arc Length**

Let’s start out with a visual that is similar to one you might see on the test:

An arc is usually represented by the symbol *s*. It is a portion of the circumference opposite a central angle. We can find out the degree of an arc by using the following equation:

*Arc length = (central angle° ÷ 360°) ** circumference*

Let’s try out the following question and solve it together:

*Refer to the diagram of circle Q, which has a radius of 8 inches. What is the length of arc AB?*

Now, we can plug the numbers into our formula:

*s = (110° ÷ 360°) ** 2**π**(8)*

*s = .306 ** 50.265*

*s = 15.38*

The length of arc *AB* is equal to 15.38 inches.

And that’s some basic info about the Math: Advanced Algebra and Functions Placement Test.

**Directions**

Choose the best answer. Use paper if necessary.

## Mathematics Advanced Practice Questions

### Question 1

Which graph is not a function?

### Question 2

Refer to the image below.

Which of the following options correctly describes Angle Q and Angle Y?

- Corresponding angles
- Alternate exterior angles
- Same-side exterior angles
- Alternate interior angles

### Question 3

All of the following angle pairs found on a transversal are always congruent except

- Corresponding angles
- Alternate interior angles
- Alternate exterior angles
- Same-side interior angles

### Question 4

Which of the following equations is equivalent to y = -1/4(x–3)² + 5?

- 4y + 20 = x² + 9
- 4y = -(x² – 6x) + 11
- 4y = (x-3)² – 5
- 4y + 5 = x² + 9

### Question 5

Simplify: (4x²y³z⁻³)² / (8x³y⁵z⁻⁴)

- z / 2xy²
- 8xy / z²
- 2xy / z²
- 2z² / xy

### Question 6

Simplify the equation.

5√2 + 3√2

- 8
- √10 + √6
- 8√2
- 8√4

### Question 7

A circle has a radius of 19 ft. What is the length of ARC AB, which central angle has a measure of 60°?

- 1.140 ft.
- 1.989 ft.
- 3.167 ft.
- 6.333 ft.

### Question 8

Solve the equation: 2x² + x – 28 = 0.

- x = -4, 3.5
- x = -3.5, 4
- x = 3.5, 4
- x = -4, -3.5

### Question 9

Solve the equation: x² – 11x + 24 = 0.

- x = -3, 8
- x = 3, 8
- x = -8, 3
- x = -8, -3

### Question 10

Triangle ABC undergoes a 90° counterclockwise rotation centered at B. What are the coordinates of the image of A under this transformation?

- (1, 7)
- (-1, 1)
- (7, 1)
- (1, -1)