California High School Proficiency Exam (CHSPE)

Preparing to take the California High School Proficiency Exam (CHSPE)?

Awesome!

You’ve found the right page. We will answer every question you have and tell you exactly what you need to study to pass the CHSPE.

Quick Facts

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

Overview

The CHSPE is for students who want to earn the legal equivalent of a high school diploma in order to enter the workforce or prepare for technical school, community college, or entrance to a university. 

The California State Board of Education awards a Certificate of Proficiency to students who pass both sections of the exam.

Format

Let’s look at an overview of what you can expect on test day.

English Language Arts Section

Mathematics Section

Screenshots and information obtained from https://www.chspe.net.

The English Language Arts section has two subtests: Reading and Language. The Reading subtest contains 84 multiple choice questions. The Language subtest contains 48 multiple choice questions and a writing task.

The Mathematics section has 50 questions, all of which are multiple choice.

You will be given 3.5 hours to take the exam. Since the individual sections are not timed, you may spend your time taking the exam however you would like. This means that you can spend more time on the areas which you find more difficult.

Cost

If you meet the Regular Registration Deadline, it will cost you $130 to take the CHSPE.

If you meet the Late Registration Deadline, the fee to take the exam is $155.00, and the fee to take the exam by the Emergency Registration Deadline is $180.00.

So, how do you register for the CHSPE? Simply go to the CHSPE site: https://www.chspe.net/registration/. You can fill out the registration form there.

Score Range

Typically, scaled scores on the CHSPE range from 250 to 400 for each section. The English Language Arts section and the Mathematics section are scored separately.

Passing Scores

In order to receive your Certificate of Proficiency, you must pass both the English Language Arts section and the Mathematics section. In order to pass the English Language Arts section, you must pass both the Language subtest and the Reading subtest.

The chart below gives you the CHPSE score breakdown and shows what you will need to make on each section.

If you score lower than a 2.5 on your writing test, you are not able to pass the English Language Arts section.

Study Time

Every student is an individual. You may need to spend either more or less time studying for the exam than the average person; it depends on how familiar you are with the content on the CHSPE.

You can use our materials to determine what content areas you struggle with the most. Then, plan to spend the majority of your studying time focusing on those areas.

As with most large exams, you cannot cram for the CHSPE. Plan to begin studying a few months before the exam and spend time studying each day.

What test takers wish they would’ve known:

  • The writing task is very important. Remember, if you do not score a 2.5 or higher on the task, you cannot pass the English Language Arts section and, therefore, you cannot earn your Certificate of Proficiency.
  • Taking practice tests can help you reduce your test anxiety.
  • You may bring a calculator to use on the Mathematics section, but it must be a nonscientific, non-programmable calculator.
  • Arrive a few minutes early to your testing center. If you arrive after the test has begun, you will not be allowed to take the exam, even if you are just 1 or 2 minutes late.
  • Get plenty of rest the night before. Do not stay up late studying. If you plan to begin studying well before the exam, you will have plenty of time to study. No one performs well on a test when they feel tired and stressed.
  • Wear comfortable clothing and bring a light jacket or sweater in case you need it in the testing center.
  • Your test results will be voided if your cell phone makes noise or if you communicate with other people taking the exam. Turn your cell phone off and put it away before the exam begins. The rules are taken very seriously during the exam.
  • You may retake the CHSPE, but you will be charged the exam fee again each time you take the test. It’s better to do your best the first time around!

Information and screenshots obtained from the CHSPE web site.

English Language Arts: Reading

The English Language Arts Reading subtest has 84 questions. You will not be timed on this specific section, but you must finish the entire test in 3.5 hours.

This subtest focuses on these six concepts:

  • Nonfiction Texts
  • Fiction Texts
  • Vocabulary
  • Main Idea and Details
  • Inferences
  • Literary Elements

So, let’s talk about them.

Nonfiction Texts

On the CHPSE Reading section, you can expect to see several different texts. Some of the texts will be non-fiction texts.

Purpose and Structure

A non-fiction text is a text that is true. In other words, it is not fiction. Remember that fiction = false and non-fiction = not false. The author may express an opinion you do not agree with, but that does not make the text fictional.

Non-fiction texts are often shorter than fiction texts. They are also frequently broken up into small sections. Think about encyclopedia articles; they are usually broken up into several sections. 

Lists are usually non-fiction texts as well. A recipe for chocolate cake and numbered steps to repair a bicycle would both be non-fiction texts in a list format.

Although non-fiction texts can be entertaining, most non-fiction texts are written with a different purpose. Some common purposes of non-fiction texts are to inform, to persuade, and to explain. The chart below gives some examples.

Types

Now that we have reviewed non-fiction texts a little bit, let’s discuss some types of non-fiction texts which often appear on the CHSPE:

  • Drug labels: Give information about prescription and non-prescription medications, such as dosage details
  • Maps: Give information (such as weather patterns, for example) about a certain area
  • Nutrition facts: Give information about the nutritional value and serving size of a food product
  • Recipes: Tell not only the ingredients to make something (usually food), but also the order in which to follow steps
  • Time schedules: Are usually divided into different columns and show when events occur
  • Warranties: Included on a product to show a manufacturer’s guarantee

When reading a fiction text, you can often skim to find an answer easily. For example, how much calcium is in the product described below?

  1. 260 mcg
  2. 10 mg
  3. 260 mg
  4. 240 g

You probably skimmed the label for the word “calcium,” which is a great strategy. In order to select the correct answer, C, it is also important that you pay attention to the details in the text.

If you’re unsure of an answer, always refer back to the text.

Read the next sample text and answer the following question.

Based on the recipe, when should you add the pieces of chicken?

  1. as soon as the butter is melted
  2. at the same time the milk is added
  3. after the potatoes
  4. at the same time as the peas and carrots

The correct answer is D. The information can be found at the end of Step 1. Since the question is about when to add an ingredient, not what the ingredient is, you can skip the “ingredients” section and skim the “instruction” section for the word “chicken.”

Fiction Texts

On the Reading section of the CHPSE, you can expect to see fiction texts as well as nonfiction texts. Let’s review some information about fiction texts now.

Purpose and Structure

Unlike a non-fiction text, a fiction text describes events that do not really happen. Sometimes, a fiction text is about something that could happen, though. If you read a realistic story in which a man walked to an ice cream shop and ordered a scoop of strawberry ice cream, it would still be a fiction story if it did not actually happen.

Fiction is often structured in a narrative format. In other words, it tells a story. It is rare to encounter a fiction text structured as a list, for example. While fiction can be used to inform, explain, or persuade, it is usually written to entertain.

Most fiction has a clear beginning, middle, and end. In other words, it usually has an obvious series of events and comes to some conclusion as the activity winds down. 

Types

Most of the time when we think of fiction, we think of examples of prose. Prose is basically any text that is not poetry. So, most non-fiction and most fiction works are prose.

Examples of prose: A pamphlet about a state park, a chapter in an autobiography

Examples of fictional prose: A short story, a chapter in a novel, a play by William Shakespeare

Unlike prose, poetry is written in verses, and while prose is broken into paragraphs, poetry is broken into stanzas. The chart below can help you to review the differences between these two types of texts.

Soon, we’ll look at some other, longer examples of fiction works and analyze their elements in more depth.

Vocabulary

On the Reading section, you will need to use your vocabulary skills. While adding to your knowledge of specific vocabulary words will help you out on the exam, related skills, such as using context clues, are important to practice before the test.

Context Clues

Context clues can help you to determine the meanings of unknown words and phrases in a text. By using context clues, you can use the words surrounding a word to learn what it means. Why is it so important to understand context clues?

On the test, you will not have a dictionary, and before the test, you will not know which words you will encounter on the exam. Using context clues is a handy skill that you can take with you wherever you go, even if you can’t bring a dictionary with you! Remember, if you don’t understand some of the words in a text, you cannot fully understand the text.

Let’s practice with an example text and question. Read the following passage:

Dantès asked to be removed from his present dungeon into another, even if it were darker and deeper, for a change, however disadvantageous, was still a change, and would afford him some amusement. He entreated to be allowed to walk about, to have fresh air, books, and writing materials. His requests were not granted, but he went on asking all the same. He accustomed himself to speaking to the new jailer, although the latter was, if possible, more taciturn than the old one; but still, to speak to a man, even though mute, was something. Dantès spoke for the sake of hearing his own voice; he had tried to speak when alone, but the sound of his voice terrified him.

Source: The Count of Monte Cristo by Alexandre Dumas (Public Domain)

What is the meaning of taciturn, as it is used in the text?

  1. strange
  2. reluctant
  3. not talkative
  4. extremely unkind

Even if you are unfamiliar with the word taciturn, you can determine that C is the correct answer by using the context clues in the passage. Taciturn is used to describe the jailer, who is also described as mute. Someone who is mute does not speak.

Take a look at the following text:

At three o’clock precisely I was at Baker Street, but Holmes had not yet returned. The landlady informed me that he had left the house shortly after eight o’clock in the morning. I sat down beside the fire, however, with the intention of awaiting him, however long he might be. I was already deeply interested in his inquiry, for, though it was surrounded by none of the grim and strange features which were associated with the two crimes which I have already recorded, still, the nature of the case and the exalted station of his client gave it a character of its own. Indeed, apart from the nature of the investigation which my friend had on hand, there was something in his masterly grasp of a situation, and his keen, incisive reasoning, which made it a pleasure to me to study his system of work, and to follow the quick, subtle methods by which he disentangled the most inextricable mysteries. So accustomed was I to his invariable success that the very possibility of his failing had ceased to enter into my head.

Source: The Adventures of Sherlock Holmes by Sir Arthur Conan Doyle (Public Domain)

Take a look at the highlighted word awaiting. This word can have different meanings depending upon how it is used. However, the underlined text “however long he might be” is a context clue. It tells you that the speaker intends to wait around for someone to arrive.

Now, look at the highlighted word inextricable. The underlined word disentangled tells you its meaning. Disentangled is the opposite of entangled, just as dissatisfied is the opposite of satisfied

The speaker is impressed that Holmes has disentangled mysteries, therefore those mysteries must be very difficult to disentangle, the meaning of inextricable.

Synonyms and Antonyms

Synonyms are words which are very similar in meaning. For example, small and little are synonyms.

Antonyms are words which are nearly opposite in meaning. Small and big are antonyms for one another.

It is important that you understand synonyms and antonyms for two reasons: you may be asked directly about synonyms and antonyms while taking the test, and synonyms and antonyms can help you determine the meanings of words. Take a look at the two examples below:

Antonym clue: We did not know that Greg was gregarious until we invited him to the party. He is rather unsociable at work, but he spoke to a lot of people on Saturday night.

Gregarious and unsociable are antonyms. Unsociable can help you understand that gregarious means sociable. 

Synonym clue: That otiose sign is entirely ineffective. Why would the restaurant manager choose to display something so purposeless?

Purposeless and ineffective are synonyms for otiose.

If you come across an unknown word in the text, a synonym or an antonym can help you decode it!

Multiple-Meaning Words

You probably use multiple-meaning words every day without realizing it. The word can is an example. Depending on how it is used, can can mean “able to,” “may,” “a storage vessel which is usually metallic,” and “to put in a can.”

While taking the Reading test, it is important to know the author’s intended meaning of the words in the text. Try this example question:

Ever since Marc and Devin had a disagreement about the software problem, they have been acting very cold toward one another.

Which of the following words is the best synonym for “cold” as it is used in the text?

  1. lukewarm
  2. freezing
  3. aloof
  4. wintry

Notice that the word cold does not relate to temperature in the text; C is the correct answer. Don’t get caught by traps like freezing and wintry; make sure that you know how the author is using words.

Main Idea and Details

On the Reading test, you will need to determine the main idea of passages and locate and analyze details. Let’s review these concepts now.

Main Idea

The main idea of a text is the “big picture,” or what the text is mainly about. A simple way to determine the main idea of a text is to ask yourself, “What is this text about?,” and to answer in a few words.

Many species of parrots have been brought to the United States to live in captivity as pets. About 25 of those species are now breeding in the wild. Based on research, one of the parrot species that most commonly establishes breeding populations in the U.S. is the red-crowned Amazon.

The red-crowned Amazon is an endangered species native to Mexico, and it is now more commonly found in the wild in the U.S. than in its native population. Both logging and the pet trade have caused the red-crowned Amazon population to dwindle in its native home.

Another introduced parrot species that can be found breeding in the wild in the U.S. is the monk parakeet. Unlike the red-crowned Amazon, the monk parakeet is not endangered. The monk parakeet breeds readily not only in the U.S., but in its native habitat, South America.

The monk parakeet is considered a nuisance by some. It is estimated to cause millions of damage to urban buildings each year because it uses these structures as nesting places.

So, what was this passage mainly about? You should automatically know that the word parrots should be found in the answer as it is included in the passage several times. The author mentions problems related to the parrots and advantages experienced by the parrots, but what accounts for these problems and advantages?

The answer is that parrots are living in the United States, where they are non-native. So, the main idea is that untamed, non-native parrot species are currently living in the U.S. 

You would not say, for example, that the main idea is that the red-crowned Amazon is an endangered species. That, of course, is just a minor detail. 

Locating Details

When you understand the main idea of the text, you are able to understand why the details are there. Think about the passage you just read about parrots. The author gives the detail that monk parrots nest on man-made structures and damage them.

This detail is included because it supports the main idea by giving evidence that non-native parrot species are now in their new habitat in the U.S.

You can expect to see some questions on the test which ask you to find specific details in the text. It is also likely that you will be asked why particular details are included. Read the following passage and answer the detail questions:

The common garter snake (Thamnophis sirtalis) has by far the most extensive geographic range of any North American reptile, covering most of the continental United States from the Atlantic to the Pacific and from south of the Mexican boundary far north into Canada and southeastern Alaska. Of the several recognized subspecies, the eastern T. s. sirtalis has the most extensive range, but that of T. s. parietalis in the region between the Mississippi River and the Rocky Mountains is almost as large. The more western T. s. fitchi occurring from the Oregon and California coasts east through the northern Great Basin, has the third largest range, while the far western subspecies pickeringi, concinnus, infernalis and tetrataenia, and the Texan T. s. annectens all have relatively small ranges.

Source: Occurrence of the Garter Snake, Thamnophis sirtalis, in the Great Plains and Rocky Mountains, by Henry S. Fitch and T. Paul Maslin (public domain)

Based on the passage, which species has the broadest range?

  1. T. s. sirtalis 
  2. T. s. fitchi 
  3. T. s. Parietalis
  4. T. s. annectens

You don’t need to worry that the names of the species are in Latin and it isn’t important to be familiar with the species. According to the second sentence, T. s. sirtalis has the “most extensive range.”

The author mentions that common garter snakes can be found as far north as Canada. This detail helps to show that:

  1. garter snakes are pests.
  2. snakes generally prefer cool climates.
  3. garter snakes can be studied in many countries.
  4. garter snakes have a broad range.

In order to answer this question, think about the main idea of the passage. The passage is intended to describe where garter snake species live. 

You can eliminate A and B because pests and climates are not mentioned. Choice C also strays from the main idea. D is correct because the information is included to describe the range of garter snakes and helps readers to understand that it spans from Mexico into Canada.

Sequences

Some texts have very obvious sequences, or orders. For example, think about the recipe you read earlier which included numbered steps to show a sequence. Other texts have less obvious sequences, but it is still important to understand the order in which events occur. 

You can use details in the text to understand a sequence of events. Look for words and phrases which relate to time. Here are a few examples:

  • “Then…”
  • “Next…”
  • “One day later…”
  • “Afterwards…”
  • “When this happened…”
  • “As soon as this occurred…” 

Take a look at the passage below and answer the question:

Tarzan had long since reached a decision as to what his future procedure should be. He had had time to recollect all that he had read of the ways of men and women in the books at the cabin. He would act as he imagined the men in the books would have acted were they in his place.

Again he rose and went into the trees, but first he tried to explain by means of signs that he would return shortly, and he did so well that Jane understood and was not afraid when he had gone.

Only a feeling of loneliness came over her and she watched the point where he had disappeared, with longing eyes, awaiting his return. As before, she was appraised of his presence by a soft sound behind her, and turned to see him coming across the turf with a great armful of branches.

Then he went back again into the jungle and in a few minutes reappeared with a quantity of soft grasses and ferns.

Two more trips he made until he had quite a pile of material at hand.

Then he spread the ferns and grasses upon the ground in a soft flat bed, and above it leaned many branches together so that they met a few feet over its center. Upon these he spread layers of huge leaves of the great elephant’s ear, and with more branches and more leaves he closed one end of the little shelter he had built.

Source: Tarzan of the Apes by Edgar Rice Burroughs (public domain)

Tarzan tries to communicate with Jane by making signs:

  1. as he returns to the shelter.
  2. as he builds a shelter.
  3. before he goes into the trees.
  4. after he goes into the trees.

The correct answer to this question is C. The words “but first” are included in the second paragraph to show that Tarzan tries to communicate with signs before going into the trees.

Cause and Effect

Many texts have a cause and effect structure. As you encounter passages on the Reading test, pay attention to what causes events to happen or characters to change.

Sometimes, a passage may use phrases such as “because of” or “as a result of” to show a cause and effect relationship, but this is not always the case. When asked a question about why something happens in the text, you may need to go back and refer to the text to find the answer.

Sometimes, it is helpful to map passages for this reason. To map a passage, you would simply write down a few words to describe each paragraph. Then, you can refer to your map in order to know which part of the text to reread to find an answer.

Let’s practice identifying causes and effects right now. Read the following passage and the question which accompanies it.

The magnificent diamond locket which hung about Tarzan’s neck, had been a source of much wonderment to Jane. She pointed to it now, and Tarzan removed it and handed the pretty bauble to her.

She saw that it was the work of a skilled artisan and that the diamonds were of great brilliancy and superbly set, but the cutting of them denoted that they were of a former day. She noticed too that the locket opened, and, pressing the hidden clasp, she saw the two halves spring apart to reveal in either section an ivory miniature.

One was of a beautiful woman and the other might have been a likeness of the man who sat beside her, except for a subtle difference of expression that was scarcely definable.

She looked up at Tarzan to find him leaning toward her gazing on the miniatures with an expression of astonishment. He reached out his hand for the locket and took it away from her, examining the likenesses within with unmistakable signs of surprise and new interest. His manner clearly denoted that he had never before seen them, nor imagined that the locket opened.

Source: Tarzan of the Apes by Edgar Rice Burroughs (public domain)

What causes Jane to become astonished?

  1. Jane is impressed by the diamonds on the locket.
  2. Jane is surprised to see the images of people in the locket.
  3. Jane cannot understand why Tarzan has never opened the locket.
  4. Jane cannot believe that Tarzan owns something pretty.

So, the effect is Jane’s astonishment. You can go back in the text and find that Jane makes “an expression of astonishment.” Immediately before this, she sees the images of the people in the locket. You can make an inference that images are responsible for her surprise.

Inferences

When you make an inference, you come to a conclusion about something that is not directly stated in the text. Good readers make inferences which are based on the information included in the text. Don’t jump to conclusions that are not supported in the text.

Drawing Conclusions

Let’s practice drawing reasonable conclusions now. Read the following passage:

Tom appeared on the sidewalk with a bucket of whitewash and a long-handled brush. He surveyed the fence, and all gladness left him and a deep melancholy settled down upon his spirit. Thirty yards of board fence nine feet high. Life to him seemed hollow, and existence but a burden. Sighing, he dipped his brush and passed it along the topmost plank; repeated the operation; did it again; compared the insignificant whitewashed streak with the far-reaching continent of unwhitewashed fence, and sat down on a tree-box discouraged.

Source: The Adventures of Tom Sawyer by Mark Twain (public domain)

When you read the passage, you can infer that Tom does not want to paint the fence with whitewash. This is because he sighs and is described as “discouraged,” furthermore, his “gladness left him and a deep melancholy settled down upon his spirit” when he looked at the fence.

Now, read the next passage and answer the question which follows it:

There are many reasons why thoughtful, cleanly, humane people should not feed upon animals, but there is a surprising deafness to this fact shown by the majority of those active in humane charities. One marvels to see hundreds of consecrated workers in session, putting forth every effort for the enacting of laws for the amelioration of the sufferings of cattle travelling to slaughter by car and ship, who are still content to patronise the butcher shop to buy food supplied by the dead bodies of these tortured victims of a false appetite. 

It is well to write, and legislate, and pray for better and kinder treatment of these frightened, thirst-maddened, tortured creatures on their journey to our tables, but the surest, quickest way to help (and this can be done even while continuing to work for the alleviation of their sufferings) is to stop feeding upon them.

The favorite phrase, “our four-footed friends,” seems rather an anachronism in the face of our acknowledged relations to them as eater and eaten, for the phrase indicates a mutual pact of friendship, which, however well sustained by them, is dishonoured by man; for even cannibals, we are told, sink no lower than to eat their foes.

Adapted from The Golden Rule Cook Book by Maud Russell Lorraine Sharpe (public domain)

Based on the text, which of the following statements is correct?

  1. The author believes that butcher shops are overpriced.
  2. The author is a vegetarian.
  3. The author dislikes the meat industry because she grew up on a farm.
  4. The author is an enthusiastic animal owner.

Choice B is the most reasonable answer. The passage focuses on the author’s disgust at the idea of eating meat. Choices A, B, and D are not supported by the text. 

Generalizing Beyond the Text

Often, you can apply ideas and events in the text to situations and concepts outside of the text. Basically, you are expanding upon the author’s ideas. Let’s make some generalizations based on the following passage:

The ship’s crew believed John Marshall to be guilty of the crime reported on Wednesday night. That evening at 10 p.m., Kate White, a guest in cabin 17, reported that after returning to her room, she saw that the door was ajar. Earlier that night, White had spoken with Marshall about the valuable necklace, which he expressed great interest in. 

White felt suspicious of Marshall, so she checked the drawer in which she kept the necklace as soon as she realized that her room was unlocked. The necklace had disappeared.

Crew members interviewed Marshall, who claimed to have been in the ship’s casino all evening. However, no guests reported having seen a man fitting Marshall’s description in the casino that night. Marshall voluntarily consented to a search of his room, in which a ladies’ diamond necklace was found.

Just as the crew decided that they had solved the case, White appeared on the scene, apologizing. White had forgotten that she had moved her necklace from its usual drawer to her luggage just before dinner. When she opened her luggage at 11:00 p.m. to find her toothbrush, she saw the necklace.

So, we can make these generalizations:

  • Marshall and White own similar pieces of jewelry.
  • White was not responsible for the incident involving the missing necklace.
  • Just because evidence suggests that a person is guilty, it does not mean that the person is necessarily guilty.
  • Sometimes people misplace their personal belongings.

When making generalizations, don’t stray too far from the text. For example, these generalizations are not supported by the text:

  • White intended to falsely accuse Marshall.
  • The necklace in Marshall’s cabin was stolen from another passenger.
  • Marshall was telling the truth about being in the casino.

We don’t know whether White meant to falsely accuse Marshall or if it was just an honest mistake. We also do not have any proof that the necklace in Marshall’s possession was stolen. Furthermore, Marshall may have been lying about visiting the casino. Maybe he was in the bathroom all evening because he was sea-sick and he was too embarrassed to tell the crew!

Making Predictions

You can make predictions about what will happen in the text based on patterns. Based on what has already happened, you can usually determine what will happen next in a story. Let’s look at an example:

Ming was a farmer who lived in the plains near the Great Wall of China many centuries ago. One morning, Ming set out to sell his plums in the market. As he walked along the path, Ming slipped in a muddy puddle and all but one of his plums tumbled from the basket he was carrying and were ruined by the mud. Ming decided that he had remarkable luck; although he had tripped, he had fallen onto soft mud instead of hard rock.

Ming continued down the road for a time until he came upon a peddler selling beautiful silk fans. Because the peddler was thirsty, Ming gave him the plum. In return, the peddler gave Ming one of the exquisite fans. Ming thanked the man and continued onward, thinking he was lucky to have traded a mere plum for such a beautiful possession.

Soon, Ming came across a group of servants carrying a wealthy merchant’s wife on a litter. When the merchant’s wife saw Ming’s fan, she called out to him because the day was growing very hot and she had forgotten to bring her own fan on her outing. Ming offered the fan to the lady, who was so grateful to him that she gave Ming her fine jade bracelet.

As Ming neared the market, he saw a horse breeder riding a rare and handsome horse. A second elegant and strong horse, tethered to a leather lead, walked behind the horse breeder. 

“Is the market busy today?” Ming asked the horse breeder.

“Yes, but I was unable to find a suitable gift for my wife, who has fine taste and enjoys jewelry” the horse breeder replied.

Adapted from a Chinese folktale

If the text were to stop here, what prediction would you make about what will happen next? You would probably predict that Ming will give the bracelet to the horse breeder who, in return, will give Ming the horse. 

This is a good prediction because it relies upon the pattern of the story. The pattern is that whenever Ming meets someone, he trades a possession for something even better. Since the horse breeder’s wife likes jewelry, you can infer that the bracelet would be a good gift for the breeder to give to his wife.

Literary Elements

When reading literature, you encounter several literary elements, such as plot and characters. You can expect to answer questions about literary elements as you complete the Reading test.

Genres

More than likely, you will see different genres of literature on the test. Remember that a genre is just a category that can be used to label a work of literature based on its style.

Here are some commonly encountered genres:

  • Action: Action works tend to be fast-paced. In other words, many events happen during a short period of time. These texts are usually described as exciting and the main character usually faces risky situations.
  • Fantasy: Fantasy works have impossible elements. For example, they may take place in a made up world or include characters like fairies or dragons. Characters in fantasy books often have supernatural abilities such as performing magic, reading minds, or knowing the future.
  • Historical Fiction: Historical fiction texts are set in the past. They may contain events that really happened and even people who really existed. However, since these texts are fictional, much of the content is invented by the author. A fiction story about a boy who fights in the American Civil War is an example.
  • Horror: Works in this genre have frightening or creepy elements. Horror texts often include ghosts, monsters, witches, or evil characters.
  • Realistic Fiction: A realistic fiction book has a contemporary setting and contains events that could really happen.
  • Science Fiction: Usually science fiction works take place in the future and focus on technology. Often these works are set on a different planet.

Practice identifying genre by matching each example description to the correct genre type. Then, check your answers.

The answers are: 1. Historical fiction, 2. Horror, 3. Science fiction, 4. Fantasy, 5. Action, and 6. Realistic fiction. Did you get all of these correct?

Plot

Plot is used to describe the events which take place over the course of the story. You have probably seen a plot diagram like the one below. Review the diagram and read the descriptions.

  • Exposition: This is the beginning of the story; the main characters are introduced and the setting (where and when the story takes place) is established.
  • Rising Action: This section builds up to the climax of the story. During the rising action, characters experience conflicts (problems).
  • Climax: The climax of the story contains the main event. It is usually the most exciting part of the story and during the climax, the main characters face a major conflict.
  • Falling Action: During this section, the action and excitement begin to wind down. 
  • Resolution: This section is also called the denouement. Most conflicts become resolved by the denouement, which usually gives the reader a sense of completion.

Think about the fairytale Goldilocks and the Three Bears. If asked to identify the rising action, for example, you could say that Goldilocks finding the bears’ home and trying their furniture and porridge are parts of the rising action. When the bears come home and find Goldilocks, the story reaches its climax.

Characters

When you read a work of literature, you usually encounter three types of characters: protagonists, antagonists, and background characters.

The protagonist is the main character or characters in the text. The antagonist opposes the protagonist(s) and creates conflict. Background characters are the other characters in the story.

Characterization is used to describe the traits of a character. These may be physical traits or other traits. Here are some examples of character traits: shy, tall, studious, curious, kind-hearted, brave, humble, and young. Round characters have a lot of character traits and are similar to people in real life. Flat characters are not described in as much detail.

Static characters stay mostly the same over the course of a text and dramatic characters change in meaningful ways. For example, a protagonist who is cowardly at the beginning of the story, but is heroic at the end of the story, is a dramatic character.

Conflict

Remember that conflict in literature is a problem experienced by a character of characters. There are different types of conflicts. For example, internal conflicts take place within a character’s mind and external conflicts occur outside of a character.

Use the chart below to review some types of conflicts as well as examples.

In order to identify the conflict, just ask yourself what is causing the problem in the story. Is it another character? A force of nature?

Many conflicts may be introduced over the course of a story. As these conflicts appear, they add interest and help to move the plot forward.

Point of View

Point of view is the perspective from which a story is told or ideas are shared. Read more about perspectives in literature:

  • First person: Events are told by someone in the text. First person perspective tends to use words like I, me, and my.
  • Second person: The speaker addresses the audience directly and uses words like you and your.
  • Third person: The narrator is outside of the story. A third-person omniscient narrator is all-knowing and may share the thoughts of many characters and events which are occurring simultaneously but in different locations. A third-person limited narrator is limited to the thoughts of just one character.

The point of view affects how the reader interprets events. For example, if a passage about an argument is told from the omniscient perspective, the reader is more likely to feel sympathy towards all of the characters involved. This is because the omniscient perspective allows you to consider the experiences of many characters.

Literary Devices

The passages on the Reading test tend to contain several different literary devices. Literary devices contribute to the style of the story and relate to both the language of the text and the reader’s experience. 

Let’s review some common literary devices and examples now.

Read the sample text and answer the following question:

The Russian gentleman was so delighted with the strawberries that the three racked their brains to find some other surprise for him. But all the racking did not bring out any idea more novel than wild cherries. And this idea occurred to them the next morning. They had seen the blossoms on the trees in the spring, and they knew where to look for wild cherries now that cherry time was here. The trees grew all up and along the rocky face of the cliff out of which the mouth of the tunnel opened. There were all sorts of trees there, birches and beeches and baby oaks and hazels, and among them, the cherry blossoms had shone like snow and silver.

Adapted from The Railway Children by Edith Nesbit (public domain)

In the final line, the author uses a simile containing _____ to help the reader imagine the cherry blossoms.

  1. personification
  2. foreshadowing
  3. alliteration
  4. the word “as”

The correct answer is C because the simile “like snow and silver” is an example of alliteration because it repeats the consonant sound s.

And that’s some basic information about the Reading test!

English Language Arts: Language

The Language subtest has 48 multiple choice questions and 1 writing task. You will not be timed on this specific section, but you must remember that you must finish the entire test in 3.5 hours.

This section can be neatly divided into 4 these main concepts:

  • Punctuation and Capitalization
  • Grammar
  • Sentence Structure
  • Writing

So, let’s talk about them.

Punctuation and Capitalization

On the test, you will be presented with sentences that improperly use punctuation and capitalization. You will need to correct those sentences. We’ll review some concepts now.

End Marks

End marks include periods (.), exclamation points (!), and question marks (?). A sentence ending in a period is meant to declare something or make a command:

Imperative (command): Please visit the Internal Revenue Service site.

Declarative (statement): I filed my tax paperwork on time this year.

An exclamatory sentence shows greater emotion than a declarative sentence:

Exclamatory: I waited on the phone for 90 minutes and I never spoke to a representative!

An interrogative sentence asks a question:

Interrogative: Why is it so difficult to speak with someone from the Internal Revenue Service?

Commas

The CHSPE Language test frequently includes items that require you to correct comma usage. Take a look at these comma rules:

Use commas to separate items in a list:

Christopher and I packed swimsuits, hiking shoes, snacks, and sunscreen.

Use a comma to separate a date and a year:

We left for the trip on May 24, 2019.

Use a comma to separate a city and a state:

The cabin that we stayed in is located in the mountains near Asheville, North Carolina.

Use a comma after introductory words and phrases:

Typically, we visit the mountains once each year.

Now that you’ve reviewed comma usage, try answering this question, which is similar to what you will see on the exam:

On our vacation, we visited a wildlife center, waded in a stream rode horses and soaked in a hot tub.

  1. we visited a wildlife center waded in a stream rode horses and soaked in a hot tub
  2. we visited a wildlife center waded in a stream rode horses and, soaked in a hot tub
  3. we visited a wildlife center, waded in a stream, rode horses and, soaked in a hot tub
  4. we visited a wildlife center, waded in a stream, rode horses, and soaked in a hot tub

The correct answer is D. The commas in this answer choice correctly separate the items in a list for clarity.

Apostrophes

Usually, students need to correct apostrophe (‘) uses on the Language test. Here are some rules for using apostrophes:

Use an apostrophe and an “s” to show possession:

The cat’s food has been spilled all over the floor.

Use an apostrophe in place of missing letters in contractions:

We can’t leave such a big mess in the kitchen.

When dealing with plural nouns, use only an apostrophe to show possession:

The Smiths’ home is always so clean.

In a case of joint ownership, the second noun takes the apostrophe:

Jane and John’s toys must be picked up before the Smiths arrive.

Now, try this practice question:

He is going to ride to Arizona in Uncle Joey’s and Aunt Cindys car.

  1. Uncle Joey’s and Aunt Cindy’s
  2. Uncle Joey and Aunt Cindy’s
  3. Uncle Joey’s and Aunt Cindys’
  4. Uncle Joeys and Aunt Cindys’

Did you choose B? If so, you are correct! If not, review Rule 4 again.

Capitalization

A word should be capitalized if it is a proper noun or if it is the first word in a sentence. Most words are also capitalized in titles. Here are a few more capitalization pointers to review:

Now, try the following example question:

She is staying at the Clearwater hotel and Spa tonight.

  1. the Clearwater Hotel and Spa
  2. The Clearwater Hotel and Spa
  3. the Clearwater hotel and spa
  4. the clearwater Hotel and Spa

Answer choice A is correct because it capitalizes words that are part of the establishment’s name, except for “and” which is a conjunction.

Grammar

In order to perform well on the Language exam, you will need to be familiar with correct grammar. Understanding standard grammar can help you to score well on both the Writing task and the sentence revision questions.

Parts of Speech

Basic knowledge of grammar includes the identification of parts of speech. Let’s review some examples now.

Now try the following practice question which relies on your knowledge of the parts of speech:

In order to prepare the soup correctly, you must pour the diced tomato mixture into the pot very slow.

  1. correctly, you must pour the diced tomato mixture into the pot very slow (no change)
  2. correct, you must pour the diced tomato mixture into the pot very slow
  3. correct, you must pour the diced tomato mixture into the pot more slow
  4. correctly, you must pour the diced tomato mixture into the pot very slowly

The correct answer is D. Correctly and slowly are adverbs that describe adjectives. Correctly describes prepare and slowly describes pour.

Pronouns and Antecedents 

Next, we’ll take a look at pronouns and antecedents. An antecedent is the noun to which a pronoun (he, she, it, they) refers. Pronouns and their antecedents must agree in number and gender. 

In the following examples, the antecedent is in bold text and the pronoun is underlined:

Incorrect: 

Like the other students, Ciara says that they would like to attend the trip.

Correct: 

Like the other students, Ciara says that she would like to attend the trip.

Incorrect:

Although the black rabbit is my favorite, none of the rabbits are aggressive, and it may be safely petted.

Correct:

Although the black rabbit is my favorite, none of the rabbits are aggressive, and they may be safely petted.

Subject/Verb Agreement

Just as pronouns must agree with their antecedents, verbs must agree with their subjects.

When the subject of a sentence is plural, the verb describing its action should also be plural. If a subject is singular, the verb which describes it should also be singular. 

Take a look at the following examples. In each example, the subject is in bold text and the verb is underlined:

Incorrect: 

Oscar, along with a few of his friends, were planning to go shopping.

Correct: 

Oscar, along with a few of his friends, was planning to go shopping.

Incorrect: 

Creativity, particularly with regard to subject matter and materials, are needed in order to win the sculpture contest.

Correct: 

Creativity, particularly with regard to subject matter and materials, is needed in order to win the sculpture contest.

Incorrect:

Each one of Dr. Brown’s students have a pencil.

Correct:

Each one of Dr. Brown’s students has a pencil.

Sentence Structure

Next, let’s take some time to think about how sentences are structured and how the structure of sentences can be improved.

Types of Sentences

When revising sentences of the CHSPE and completing the Writing task, you will need to understand how to structure sentences correctly. This knowledge will help you to avoid errors such as run-on sentences and sentence fragments.

First, it’s important to know the difference between an independent clause and a dependent clause. An independent clause can stand on its own. It is a complete thought with a subject and a verb. 

A dependent clause is not a complete thought and, therefore, cannot be a sentence on its own. It must be joined with an independent clause.

In the following examples, the independent clauses are red and the dependent clauses are blue. Notice that the dependent clauses could not be complete sentences on their own.

  • Doug says that the people who purchased the lot across the street will tear down the old house.
  • The old house is in poor condition. 
  • The people will build a new house.
  • When construction on the new house begins, this street will become even noisier. 

Now that we’ve reviewed clauses, we’re ready to look at different sentence structures:

Simple sentence: Has an independent clause with no conjunction or dependent clause.

I will visit the tailor shop today.

The tailor shop was closed yesterday.

Compound sentence: Has two independent clauses joined by a conjunction 

I will visit the tailor shop today because it was closed yesterday.

Complex sentence: Has one independent clause and at least one dependent clause. 

Now that my skirt has been mended, I must get it from the shop.

Compound-complex sentence: Has multiple independent clauses and at least one dependent clause

Now that my skirt has been mended, I will go get it, and I will thank the tailor.

Revising Sentences

On the exam, you will be asked to choose which versions of sentences are best. You will need to avoid sentence fragments, which only contain dependent clauses, and run-on sentences. A run-on sentence has more than one independent clause and the clauses are not combined with a coordinating conjunction.

In order to correct sentences on the exam, you may need to combine them with other sentences, add words to them, or break them into multiple sentences. A good way to decide which answer choice is correct is to read each option and to choose the option which sounds the most natural and clear. 

You should also choose answers which are precise. Consider this sentence:

There are extra chairs in the room, and they are not needed.

It would be better to say:

There are extra chairs in the room.

It’s best to eliminate unnecessary information and get to the point. Otherwise, the language is confusing for readers.

Let’s try a practice question now.

When we left to go to the recital the rain was pouring down so hard we could barely see anything.

  1. (No changes needed)
  2. When we left to go to the recital, the rain was pouring down so hard we could barely see anything.
  3. When we left to go to the recital, the rain was pouring down so hard, we could barely see anything.
  4. When we left to go to the recital, the rain was pouring down hard, and we could barely see.

Choice D is the correct answer because it is precise and it uses commas and conjunctions appropriately.

Writing

You might be curious about what’s on the CHPSE Writing task. You will be presented with a brief writing prompt that asks for your opinion about an issue. You will need to respond to the prompt with a persuasive essay that is a few paragraphs long.

Here is an example of a writing prompt which is similar to what you are likely to see on the test:

According to some people, classes for high school students should begin after 9:00 a.m. Do you agree or disagree with this opinion? Write an opinion letter to your local newspaper which discusses your stance and supports it with evidence. Your writing should be focused on persuading readers to agree with your opinion.

The following list contains tips for your essay:

  1. Clearly introduce your opinion, provide supporting paragraphs, and give a strong conclusion.
  2. Include a topic sentence for each paragraph to help the reader understand the point that you are trying to make.
  3. Stay on-topic as you write. Only include details that support the main idea of your essay.
  4. Plan your essay before you begin to write.
  5. Check your spelling and use vocabulary terms correctly.

Now, let’s get a little bit more specific.

Pre-writing

Pre-writing includes the thought and planning that you should apply to your essay before you begin writing it. During the pre-writing stage, you should read (and re-read) the prompt

You may be an excellent writer, but if you do not read carefully and make sure that you understand the prompt, your essay will be off-topic.

After carefully considering the prompt, brainstorm. Decide on your opinion and jot down any reasons that come to mind for why you have that opinion. Remember, you are not being judged for your beliefs. You are being scored according to how well you can support an opinion.

After determining your opinion, decide upon 3 or 4 supporting arguments or reasons to use in your essay. Choose the strongest evidence and eliminate the rest. 

Plan to have an introduction, supporting paragraphs, and a conclusion. Create a brief outline that shows what you will focus on in each paragraph. This will keep you from straying from the topic. Here is an example of an outline which might be created as a response to the example prompt you were given earlier:

  1. Paragraph 1: Introduce the topic and state opinion that class should begin before 9 a.m.
  2. Paragraph 2: Supporting paragraph – students should begin class early because they have sports and jobs after school.
  3. Paragraph 3: Supporting paragraph – starting class early prepares students for the workforce.
  4. Paragraph 4: Supporting paragraph – studies show that people remember more when they learn in the morning, as opposed to the afternoon.
  5. Final paragraph: Summarize the main points and restate the opinion in different words.

Writing

Once you have your outline, you can begin writing your response. Stick to the outline and support your opinions with details. If you can address any counter-arguments, refute them.

Make your sentences clear and precise and try to use different vocabulary words as you write. However, if you are unsure of the meaning of the word, avoid including it.

Check out the following two example responses to the prompt. The topic sentences are in bold and counter-arguments that are addressed are underlined. 

Example Prompt: According to some people, classes for high school students should begin after 9:00 a.m. Do you agree or disagree with this opinion? Write an opinion letter to your local newspaper which discusses your stance and supports it with evidence. Your writing should be focused on persuading readers to agree with your opinion.

Example Response 1 (4 score points):

Although some people believe that high school classes should begin after 9:00 a.m., I highly disagree with that stance. It is better for high school students to begin their classes earlier, rather than later, each day. Both research and logic support the idea that students should attend classes before 9:00 a.m.

First of all, many students are required to work, attend club meetings, and participate in sports after school. Students must be able to finish their classes each day in a timely manner so that they can fulfill these other duties. If classes start at 7:30 in the morning instead of at 9:00 a.m., students have an extra 90 minutes each afternoon to get to work, meetings, and practices. Some people may argue that not all students work or are involved in sports or clubs. However, all students must study and rest after school, so all students would benefit from being excused from class earlier.

Whether or not high school students currently work, they are all preparing to work in the future. By arriving to school early, students can acclimate to the routine of the average full-time employee. Most employees are required to be present at work before 9:00 a.m. By getting up early for school, students are practicing the habits of responsible employees.

Not only will students benefit in the future by getting up early for school, they will experience the positive effects of this practice right now. Studies show that students are more attentive in the earlier hours of the day and that they are more likely to retain information if they study in the morning. Since education is so crucial, it is important that students have every advantage to perform their best. Offering early classes is a great way to allow students to shine since they perform best in the morning.

Since there is so much evidence that students should begin school before 9:00 a.m., it is clearly in the best interest of these individuals to arrive at school early. Attending school early gives students time to fulfill their duties in the afternoon and it allows them to practice the habits of adult employees and to retain what they learn. Clearly, classes have historically begun early in the morning for a good reason.

Notice that this essay has a clear introduction and conclusion. The writer gives details in the body paragraphs, stays on-topic, and addresses counterarguments. There are also clear connections between the paragraphs. Now take a look at the second essay which addresses the same prompt:

Example Response 2 (2 score points):

I think very strongly that class should start after 9:00 a.m. so I agree with the point. There is no reason for students to go to school so early and my arguement is that students need time to get enough sleep so they can pay attention in school and not be sleepy. If students don’t sleep enough then they will be too tired and won’t be able to pay attention in class. If school would start later, then students would probably enjoy school a lot more. If I could choose a time for school to start it would probably be at ten. That way students can get rest and then go to class. I think plenty of people that I know would agree with my point especailly if they are students. Students would be very happy and awake if school doesn’t start until ten. That would be the best time. 

If I were in charge I would change things for students so that they would be happier and enjoy school and everything else a lot more. 

The second example response states an opinion and it provides some support for the opinion. However, it would only receive half of the score given to the first example response. The second response is vague and contains grammatical and spelling errors. The ideas are not clearly developed and the response does not have a clear structure.

Revision

After you write your response to the Writing prompt, you should take some time to revise your essay. You might be surprised at the errors you find in your own work.

Take a look at these revision tips:

Remember that the majority of your time should be spent planning and writing, not revising, but revision is still an important step.

And that’s a basic overview of the CHPSE Language test!

Mathematics

The Mathematics section has 50 multiple-choice questions. You will not be timed on this specific section, but you must remember that you must finish the entire test in 3.5 hours.

This section can be neatly divided into these 6 main concepts:

  • Number Sense and Operations
  • Patterns and Relationships
  • Algebra
  • Statistics and Probability
  • Geometry
  • Measurement

So, let’s talk about them.

Number Sense and Operations

On the test, you will need to evaluate numbers and perform basic operations. We’ll review some related concepts now.

Comparing Numbers

If you are asked to compare the value of whole numbers, such as 24 and 26, it is pretty easy to decide which one is greater. Sometimes, you may need to analyze fractions and decimals as well. This requires a bit more work, but it’s still pretty simple.

Let’s say that you are asked to list the following numbers from least to greatest: .75, 3, -3, 3/10 and 25%.

It is pretty obvious that -3 has the least value since it is negative and that 3 has the greatest value because it is the only whole, positive number. The numbers .75, 25%, and 3/10 are positive, and they are each less than 1.

We can compare them more easily by converting 25% and 3/10 to decimals. In order to convert a percent to a decimal, just move the decimal two places to the left:

25. becomes .25

To convert a fraction to a decimal, divide the numerator (top number) by the denominator (bottom number):

3 ÷ 10 = .30

Now you can easily order the numbers from least to greatest:

-3, .25, .30, .75, 3
or
-3, 25%, 3/10, .75, 3

Scientific Notation

Next, we’ll review scientific notation, which is another way to express numbers. Scientific notation uses exponents, which show how many times to multiply a number by itself. For example:

The exponents in scientific notation are always applied to the number 10. As you review the chart below, remember that any non-zero number raised to the power of 0 is 1.

Notice that since scientific notation always uses the number 10 raised to a power, you only need to move the decimal. There is no reason to perform any complicated multiplication!

Order of Operations

When solving problems on the CHPSE Mathematics test, it is important that you perform operations in the correct order. 

You may have used the acronym “PEMDAS” before. This acronym stands for Parentheses, Exponents, Multiplication/Division, Addition/Subtraction. The phrase “Please Excuse My Dear Aunt Sally” will help you to remember “PEMDAS,” which shows the correct order in which to perform operations.

Let’s practice with a sample equation and solve it step by step:

16 + (2 x 5) x 3⁴ ÷ 9

  1. Parentheses: 16 + (10) x 3⁴ ÷ 9
  2. Exponents: 16 + (10) x 81 ÷ 9
  3. Multiplication/Division: 16 + 810 ÷ 9
          = 16 + 90
  4. Addition/Subtraction: 106

Remember to solve problems from left to right. Notice that in the multiplication/division step, we multiplied (left) before we divided (right).

It’s very important to know the order of operations – this concept will come in handy for lots of problems on the test!

Absolute Value

A bar on either side of a number indicates absolute value. For example, this indicates the absolute value of 4:

|4|

The easiest way to think about absolute value is to imagine it is the number of places a value is from 0 on the number line. So, the absolute value of 4 and -4 are the same:

Likewise, the absolute value of 5 and -5 are the same, and so on.

Patterns and Relationships

On the Mathematics test, you will need to evaluate patterns by determining the relationships between numbers, symbols, and figures.

Number Patterns

Often, students must find a term in a sequence of numbers. Sometimes, you need to figure out the relationship between numbers yourself. Take the following sequence for example:

2, 4, 8, 16, ___

By considering how each term compares to the term before it, you can determine that the pattern is to multiply each term by 2 to get the next term. The next term in the sequence would be 32.

Other times, you may actually be told how to find the next term. Try answering the following question:

What is the 5th term in the sequence?

  1. .25
  2. .5
  3. .75
  4. 1

So, you are interested in the 5th term, or when n = 5. Set n to 5 in the equation:

8(0.5)⁵⁻¹ =
8(0.5) =
8(.0625) =
.5

Once again, the order of operations is important. If you had multiplied first, you would have gotten 256, which is incorrect, as your answer.

Patterns of Symbols and Figures

Not all patterns are expressed using numbers. Often, the test includes questions that ask test-takers to evaluate a sequence of symbols or geometric figures. When presented with symbols or figures in a sequence, consider how each item compares to the one before it, just as you would with numbers.

Try this example question:

  1. 1
  2. 2
  3. 3
  4. 4

Look at the first term, which shows a circle inside a square. The next term shows a square with a circle on top of it. So, it seems like the pattern is to repeat the same large shape and move the shape inside of it to the top.

We can check that pattern by looking at the third and fourth terms. The third term shows a triangle inside a square. The next term shows that the triangle has been moved to the top of the square. So far, so good–the pattern seems to be correct.

The fifth term shows a diamond inside of a triangle. In the next term, the diamond is moved to the top, but it is on a circle, not a triangle. So, the same large shape is not being repeated.

You can conclude that the only rule to the pattern is that the small shape appears inside of a large shape and then the small shape appears on top of a large shape. As long as the small shape stays the same and moves to the top, the pattern is followed. The large shape can be any shape.

Since choice C is the only choice that shows the same small shape, a heart, moved to the top of a large shape, it is correct.

Algebra

You can expect to see some basic algebra concepts on the Mathematics test and to solve and evaluate equations. We will review some algebra concepts together now.

Linear Equations

A linear equation forms a straight line when it is graphed. The standard form of a linear equation is y = mx + b. The m is the slope of the line and the y is the y-intercept. The y-intercept is the point at which y = 0 on a graph.

Here’s an example of a linear equation:

y = 2x + 1

We can plug different numbers in for x to get y. Let’s find y when x = 1:

y = 2(1) + 1
y = 2 + 1
y = 3

If x = 1, y = 3. Points are graphed in the form (x, y). So, (1, 3) is a point on the line:

By plugging 3 in for x, we find that y = 5. So, (3,5) is another point on our line:

Linear equations are not always in the standard form y = mx + b. Consider this linear equation:

5x = 6 + 3y

We can rearrange the question to get the y value alone. Then the equation will be in standard form. Remember that whatever you do on one side of the equation, you must also do on the other side. For example, we must add 4 to both sides:

Now it is easier to find the value of y given any x value.

A system of linear equations is two or more linear equations working together. Here is an example:

x + y = 6
-3x + y = 2

We can solve that system by subtracting the second equation from the first:

Now we can find the y value by plugging 1 in for x. Let’s plug the 1 into the first equation:

x + y = 6
1 + y = 6
y = 5

So, you now have the x and y values for the system of equations.

Slope 

If you do not know the slope of a line, you can find the slope with any two points on the line. Here is the formula for slope:

Now, let’s try an example question that requires us to find the slope of a line:

On a coordinate plane, at what point does a line passing through point A (5, –2) and point B (3, 4) intersect the y-axis?

Even though the question asks for the y-axis, we cannot find that value until we find the slope. Plug the given values into the slope formula. Whichever point you chose for y₁ should also be used for x₁.

Now we can plug the slope and the values from either point into a linear equation. Let’s use the values from point A.

y = mx + b
4 = (-3)3 + b
4 = -9 + b
13 = b

So, the y-intercept of the line is 13.

Quadratic Equations

Instead of forming a straight line when graphed, a quadratic equation forms a curve, which is called a parabola. The standard form of a quadratic equation is ax² + bx + c = 0. 

Just like linear equations, quadratic equations are not always in standard form. For example:

2a² = 16

We can rearrange this quadratic equation so that it is in standard form by subtracting 16 from each side of the equation to get:

2a² – 16 = 0

You can solve a quadratic equation by using the quadratic formula:

Notice the root symbol in the formula. It is telling you to find the square root. The square root of a number is the number which multiplied by itself equals the original number. For example, the square root of 16 is 4 because 4 x 4 = 16. 

Let’s solve the quadratic equation + 4x – 14 = 0 by plugging it into the quadratic formula:

Unlike a linear equation, a quadratic equation does not have just one answer. The plus or minus sign (∓) says that we can add or subtract. Depending on whether we add or subtract, the outcome is different.

Geometry

Next, we’ll talk a little bit about geometry. On the exam, you can expect to answer geometry questions about concepts like perimeter and area. Let’s review.

Perimeter

Perimeter is the measurement of the outer edges of a geometric figure, such as a triangle. The following table gives the formulas to find the perimeter of figures, as well as some examples:

So, let’s say that you have a rectangular piece of paper that measures 8 inches by 6 inches. Add all of the sides to find the perimeter:

8 + 8 + 6 + 6 = 28

The perimeter of the paper is 28 inches.

Area

Area is expressed in square units (²). The area describes the space inside a figure. The following table gives formulas for area and examples.

Let’s practice finding the area of a figure.

If the height of a triangle is 12 inches and the base is 20 inches, what is the area of the triangle?

a = ½(bh)
a = ½(12 x 20)
a = ½(240)
a = 120

So, the area of the triangle is 120 in.²!

Circles

You can expect to answer some questions about circles on the test as well. The formula for the circumference of a circle is c = 2πr. Circumference is the measurement around the circle. The area of a circle can be found using the formula a = πr².

In these formulas, r indicates the radius. A circle’s diameter is the length of a line from one side of the circle to the other that passes through the center. The radius is half of the diameter.

Let’s look at an example and find the circumference and the area:

c = 2πr
c = 2π8
c = 50.27 ft.

a = πr²
a = π8²
a = 201.06 ft.²

Since you know the radius, it is easy to find both measurements. Just remember the formulas!

Angles

If lines are perpendicular, they create right angles, which equals 90°. Lines that are parallel to each other never intersect. 

A transversal is a line that crosses two parallel lines. Transversals create several different angles:

The angles you see in the diagram can be labeled as follows:

  • Exterior angles: ∠A∠F∠G∠D 
  • Interior angles: ∠B∠E∠H∠C
  • Consecutive interior angle pairs: [∠B and∠E] [∠H and∠C]
  • Alternate exterior angle pairs: [∠A and∠G] [∠Fand∠D]
  • Alternate interior angle pairs: [∠E and∠C] [∠H and∠B]
  • Corresponding angle pairs: [∠A and∠E] [∠C and∠G] [∠D and∠H] [∠F and∠B] 

Since a line is always equal to 180°, you can use the value of one angle to find the values of other angles. 

For example, if ∠A is equal to 140°, you know that ∠E is also equal to 140° and ∠B and ∠F are equal to 40°.

Volume

Just as area described the space inside a two-dimensional figure, volume describes the space inside a three-dimensional object. Review the three-dimensional objects below.

Here are the formulas to find the volume of each object shown in the diagram:

  • Cube: v = lwh
  • Cylinder: v = πr²h
  • Rectangular pyramid: v = (lwh) / 3

So, let’s practice by finding the volume of a cube. If you have a cube that is 3 inches tall and all of the sides are perfectly square, what is the volume of the cube?

3 x 3 x 3 = 27 in.³

By using the volume formulas, it is simple to find the volume of the objects when given the appropriate variables (such as height). All you need to do is plug in the numbers.

Measurement

On the CHSPE, you will most likely be presented with word problems and visuals that relate to measurement. We’ll review some concepts and practice now.

Time, Rate, and Distance

If you say that a car is traveling 60 mph (miles per hour), you are describing its rate. In an hour (time), the car travels 60 miles (distance). In the formulas below, r = rate, d = distance, and t = time.

  • r = d ÷ t
  • d = rt
  • t = d ÷ r

Let’s practice using one of the formulas:

If a car travels at a constant speed of 45 mph, how long does it take the car to travel 20 miles?

  1. 2 hours and 15 minutes
  2. about an hour
  3. 3 hours
  4. about 30 minutes

The question gives you the rate and the distance. Plug these values into the formula for time to get 20 ÷ 45 = 0.444. Since 0.444 of an hour is about 30 minutes, D is the best answer.

Estimation

The last question asked you to give an approximate value. Often, the Mathematics test asks test-takers to make estimates and give approximations. An estimate does not need to be perfect, it just needs to be the best answer.

Here is an example question which is similar to what you may see on the test:

Dottie is planning to put up a small wire fence to keep rabbits out of her garden. The image below shows a drawing of the garden. About how many feet of wire fencing material does she need to surround the garden?

  1. 400 ft.
  2. 40 ft.
  3. 60 ft.
  4. 100 ft.

This question is asking you to find the perimeter, but you only know the length of one side. You will need to estimate the length of the other sides. The longer sides are about twice the length of the shorter sides.

Perimeter = 20 + 20 + 10 + 10 = 60

Option C is the best response. You can eliminate some answers immediately. For example, 40 ft. is obviously not right because it only accounts for two sides of the garden.

And that’s some basic info about the CHPSE.