- Order of operations (PEMDAS)
- Parentheses, Exponents, Multiplication/Division (left→right), Addition/Subtraction (left→right).
- Slope formula
- Slope = rise ÷ run = (y₂ − y₁) ÷ (x₂ − x₁); the m in y = mx + b.
- Slope-intercept form
- y = mx + b, where m is the slope and b is the y-intercept (where the line crosses the y-axis).
- Y-intercept
- The point where a line crosses the y-axis (where x = 0); the value b in y = mx + b.
- X-intercept
- The point where a line crosses the x-axis (where y = 0); set y = 0 and solve for x.
- Linear equation
- An equation whose graph is a straight line; variables appear only to the first power, e.g. y = 2x + 1.
- Solving a one-variable equation
- Isolate the variable by doing the same inverse operation to both sides until the variable stands alone.
- Inequality sign flip rule
- When you multiply or divide both sides of an inequality by a negative number, reverse the inequality sign.
- Literal equation
- An equation with several variables; 'solve for' one variable by isolating it in terms of the others (e.g. solve A = lw for w).
- Quadratic equation
- An equation of the form ax² + bx + c = 0; its graph is a parabola. Solve by factoring, the square root, or the quadratic formula.
- Quadratic formula
- x = (−b ± √(b² − 4ac)) ÷ (2a), which solves any quadratic ax² + bx + c = 0.
- Discriminant
- b² − 4ac inside the quadratic formula; positive → 2 real roots, zero → 1, negative → no real roots.
- Factoring a quadratic
- Rewrite ax² + bx + c as a product of two binomials; the roots are the values that make each factor 0.
- FOIL
- Multiply two binomials by combining First, Outer, Inner, Last terms: (x + 2)(x + 3) = x² + 5x + 6.
- Difference of squares
- a² − b² = (a + b)(a − b); a common PERT factoring pattern, e.g. x² − 9 = (x + 3)(x − 3).
- Polynomial
- A sum of terms, each a number times a variable raised to a whole-number power, e.g. 3x² − 2x + 5.
- Like terms
- Terms with the same variable and exponent; only like terms can be combined (3x + 5x = 8x).
- Distributive property
- a(b + c) = ab + ac; multiply the outside factor by each term inside the parentheses.
- System of equations
- Two or more equations solved together; solve by substitution or elimination to find the values that satisfy all of them.
- Substitution method
- Solve one equation for a variable, then substitute that expression into the other equation.
- Elimination method
- Add or subtract the equations so one variable cancels, then solve for the remaining variable.
- Function
- A rule that assigns exactly one output to each input; written f(x), where f(x) is the output for input x.
- Evaluating a function
- Substitute the input value for x: if f(x) = 2x + 1, then f(3) = 2(3) + 1 = 7.
- Domain of a function
- The set of all allowable input (x) values for the function.
- Range of a function
- The set of all output (y) values the function produces.
- Exponent product rule
- xᵃ · xᵇ = xᵃ⁺ᵇ — add exponents when multiplying powers with the same base.
- Exponent quotient rule
- xᵃ ÷ xᵇ = xᵃ⁻ᵇ — subtract exponents when dividing powers with the same base.
- Power of a power
- (xᵃ)ᵇ = xᵃᵇ — multiply the exponents.
- Zero exponent
- Any nonzero number raised to the 0 power equals 1 (x⁰ = 1).
- Negative exponent
- x⁻ⁿ = 1 ÷ xⁿ — a negative exponent means the reciprocal.
- Square root
- A value that, multiplied by itself, gives the number; √25 = 5. The radical √ asks for the principal (positive) root.
- Rational number
- A number that can be written as a fraction of two integers (e.g. ½, 0.75, −3).
- Irrational number
- A number that cannot be written as a fraction; its decimal never ends or repeats (e.g. √2, π).
- Scientific notation
- Writing a number as a digit between 1 and 10 times a power of 10, e.g. 5,300 = 5.3 × 10³.
- Absolute value
- The distance of a number from 0 on the number line, always nonnegative; |−7| = 7.
- Percent to decimal
- Divide by 100 (or move the decimal two places left): 25% = 0.25.
- Percent of a number
- Convert to a decimal and multiply: 20% of 80 = 0.20 × 80 = 16.
- Percent change
- (new − old) ÷ old × 100; positive is an increase, negative a decrease.
- Proportion
- An equation stating two ratios are equal; solve by cross-multiplying.
- Cross-multiplying
- For a/b = c/d, multiply diagonally: a·d = b·c, then solve.
- Ratio
- A comparison of two quantities by division, written a:b or a/b.
- Mean (average)
- Add all the values, then divide by how many values there are.
- Median
- The middle value of a data set arranged in order (average the two middle values if the count is even).
- Mode
- The value that appears most often in a data set.
- Range (statistics)
- The highest value minus the lowest value in a data set.
- Probability
- Favorable outcomes ÷ total outcomes, a number from 0 (impossible) to 1 (certain).
- Area of a rectangle
- Area = length × width.
- Area of a triangle
- Area = ½ × base × height.
- Area of a circle
- Area = πr², where r is the radius.
- Circumference of a circle
- Circumference = 2πr (or πd), where r is the radius and d the diameter.
- Perimeter
- The total distance around a figure — add the lengths of all its sides.
- Volume of a rectangular prism
- Volume = length × width × height.
- Volume of a cylinder
- Volume = πr²h, where r is the base radius and h the height.
- Pythagorean theorem
- In a right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
- Radius vs. diameter
- The radius is the distance from a circle's center to its edge; the diameter is twice the radius (d = 2r).
- Supplementary angles
- Two angles whose measures add to 180°.
- Complementary angles
- Two angles whose measures add to 90°.
- Vertical angles
- Opposite angles formed by two intersecting lines; they are equal.
- Coordinate plane
- A grid formed by a horizontal x-axis and vertical y-axis; points are written (x, y).
- Quadrants
- The four regions of the coordinate plane, numbered I–IV counterclockwise starting at the upper right.
- Midpoint formula
- The midpoint of two points is ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2).
- Distance formula
- Distance = √((x₂ − x₁)² + (y₂ − y₁)²); it comes from the Pythagorean theorem.
- Parallel lines
- Lines in the same plane that never meet; they have equal slopes.
- Perpendicular lines
- Lines that meet at a 90° angle; their slopes are negative reciprocals (e.g. 2 and −½).
- Simplifying fractions
- Divide numerator and denominator by their greatest common factor.
- Adding fractions
- Find a common denominator, add the numerators, then simplify.
- Multiplying fractions
- Multiply the numerators together and the denominators together, then simplify.
- Dividing fractions
- Multiply by the reciprocal of the second fraction (flip and multiply).
- Rational expression
- A fraction whose numerator and/or denominator are polynomials; simplify by factoring and canceling common factors.
- Greatest common factor (GCF)
- The largest number (or expression) that divides two values evenly.
- Least common multiple (LCM)
- The smallest number that two or more numbers both divide into evenly.
- Prime number
- A whole number greater than 1 divisible only by 1 and itself (2, 3, 5, 7, 11, …).
- Integer
- A whole number and its opposite, including 0 (…, −2, −1, 0, 1, 2, …).
- Inverse operations
- Operations that undo each other — addition/subtraction, multiplication/division, squaring/square root.
- Translating word problems
- Turn the words into an equation first; 'is' means =, 'of' means ×, 'more than' means +.
- Direct variation
- y = kx; as x increases, y increases proportionally by the constant k.
- Line of best fit
- A straight line drawn through scatter-plot data to show the trend; its slope is the rate of change.
- Mean vs. median for outliers
- The median better describes the typical value when a data set has extreme outliers.
- Solving a proportion word problem
- Set the matching quantities as equal ratios, then cross-multiply and solve for the unknown.
- Unit rate
- A rate with a denominator of 1, such as miles per hour or cost per item; divide to find it.
- Markup
- An increase added to a cost: new price = original + (markup% × original).
- Discount
- A reduction in price: sale price = original − (discount% × original), or pay (100% − discount%).
- Simple interest
- Interest = principal × rate × time (I = Prt), with the rate as a decimal and time in years.
- Compound vs. simple interest
- Simple interest earns only on the principal; compound interest earns on the principal plus prior interest.
- Estimating an answer
- Round numbers to convenient values to predict the size of the answer and catch calculator slips.
- Combining like terms
- Add or subtract the coefficients of terms with the identical variable part: 4x + 2x = 6x.
- Solving with variables on both sides
- Move all variable terms to one side and constants to the other, then isolate the variable.
- Graphing a linear inequality
- Graph the boundary line (dashed for < or >, solid for ≤ or ≥), then shade the side that satisfies it.
- Slope of a horizontal line
- Zero — a horizontal line has no rise.
- Slope of a vertical line
- Undefined — a vertical line has no run, so you would divide by zero.
- Parabola
- The U-shaped graph of a quadratic; it opens up if a > 0 and down if a < 0.
- Vertex of a parabola
- The highest or lowest point of the parabola; the x-value is −b ÷ (2a).
- Exponential growth
- A quantity that increases by a constant percentage each period, modeled by y = a·bˣ with b > 1.
- Solving radical equations
- Isolate the radical, square both sides, solve, then check for extraneous solutions.
- Rationalizing a denominator
- Eliminate a radical in the denominator by multiplying numerator and denominator by that radical.
- Mixed number to improper fraction
- Multiply the whole number by the denominator, add the numerator, and keep the denominator.
- Decimal to percent
- Multiply by 100 (move the decimal two places right): 0.4 = 40%.
- Fraction to decimal
- Divide the numerator by the denominator: ¾ = 3 ÷ 4 = 0.75.
- Reading a bar graph
- Compare the heights or lengths of bars to read and compare category values.
- Reading a line graph
- Follow the line to read how a value changes over time or another variable.
- Reading a circle (pie) graph
- Each slice is a share of the whole; the full circle represents 100%.
- Surface area
- The total area of all the outer faces of a 3-D solid, measured in square units.
- Congruent vs. similar figures
- Congruent figures have equal size and shape; similar figures have the same shape but proportional sizes.
- Main idea
- The central point a passage makes — what the whole text is mostly about.
- Supporting detail
- A fact, example, statistic, or reason that develops or proves the main idea.
- Topic sentence
- The sentence (often first) that states a paragraph's main idea.
- Author's purpose
- The reason a text was written — to inform, persuade, entertain, or explain.
- Tone
- The author's attitude toward the subject, revealed through word choice (e.g. formal, critical, hopeful).
- Mood
- The feeling a text creates in the reader, set by imagery and word choice.
- Inference
- A logical conclusion drawn from textual evidence plus reasoning — not stated outright.
- Drawing conclusions
- Combining stated details with reasoning to decide what the text implies.
- Context clues
- Surrounding words and sentences that help you figure out an unfamiliar word's meaning.
- Word meaning in context
- Determining what a word means as it is actually used in a passage — not just its dictionary definition.
- Denotation
- A word's literal, dictionary definition.
- Connotation
- The emotional or implied meaning a word carries beyond its literal definition.
- Figurative language
- Language that means more than the literal words, such as metaphor, simile, and personification.
- Simile
- A comparison using 'like' or 'as' (e.g. 'brave as a lion').
- Metaphor
- A direct comparison that states one thing is another (e.g. 'time is money').
- Personification
- Giving human qualities to something nonhuman (e.g. 'the wind whispered').
- Literary text
- Imaginative writing such as fiction, poetry, and drama.
- Informational text
- Nonfiction such as essays, articles, editorials, and documents that present facts and arguments.
- Text structure
- How a text is organized — e.g. cause/effect, compare/contrast, sequence, or problem/solution.
- Cause and effect
- A text structure that explains why something happens (cause) and what results (effect).
- Compare and contrast
- A structure that examines similarities and differences between ideas or things.
- Sequence
- A structure that presents events or steps in order.
- Problem and solution
- A structure that states an issue and then proposes a way to resolve it.
- Summarizing
- Briefly restating a passage's main points in your own words, leaving out minor details.
- Paraphrasing
- Restating a specific passage in your own words while keeping the original meaning.
- Claim
- A debatable position an author argues for.
- Evidence
- The facts, examples, data, or expert opinions an author uses to support a claim.
- Argument
- A claim backed by reasoning and evidence intended to persuade the reader.
- Counterargument
- An opposing view that an author acknowledges and responds to.
- Bias
- An author's slant or prejudice that may distort how a topic is presented.
- Fact vs. opinion
- A fact can be proven true or false; an opinion expresses a belief, judgment, or feeling.
- Point of view
- The perspective from which a text is told or an issue is presented.
- Theme
- The underlying message or central insight about life in a literary work.
- Synthesis
- Combining ideas from two or more texts to reach a broader conclusion.
- Primary source
- A firsthand or original record, such as a letter, speech, or diary.
- Secondary source
- A source that analyzes or interprets primary sources, such as a textbook or review.
- Implicit vs. explicit
- Explicit information is stated directly; implicit information is suggested and must be inferred.
- Generalization
- A broad statement drawn from specific examples; valid ones are supported by sufficient evidence.
- Persuasive techniques
- Strategies authors use to convince readers — appeals to emotion, logic, or credibility.
- Connotative vs. denotative
- Denotation is a word's literal meaning; connotation is its implied or emotional meaning.
- Vocabulary in context
- Use nearby words, examples, or contrasts in the sentence to define an unfamiliar word.
- Prefix
- A word part added to the front of a root that changes its meaning (e.g. 'un-' means not).
- Suffix
- A word part added to the end of a root, often changing its part of speech (e.g. '-ly').
- Root word
- The base part of a word that carries its core meaning.
- Analyzing structure
- Examining how a paragraph or whole text is built to understand the author's logic and emphasis.
- Drawing on evidence
- Anchoring every reading answer to specific words or lines in the passage.
- Tone words
- Descriptive words for attitude — e.g. sarcastic, optimistic, somber, objective, urgent.
- Implied main idea
- A main idea that is not stated directly; the reader infers it from the details.
- Author's craft
- The deliberate choices an author makes — word choice, structure, and devices — to shape meaning.
- Making predictions
- Using clues in a text to anticipate what will happen or be said next.
- Comparing two passages
- Identifying where paired texts agree, disagree, or emphasize different points.
- Reading a graphic
- Interpreting a chart, graph, table, or map that accompanies a passage to draw conclusions.
- Relevant vs. irrelevant detail
- Relevant details support the main idea; irrelevant ones do not and can be set aside.
- Audience
- The intended readers a text is written for, which shapes its tone, word choice, and content.
- Key ideas and details
- A PERT Reading skill area: identifying main ideas, supporting details, and what the text states or implies.
- Craft and structure
- A PERT Reading skill area: how word choice, text structure, and point of view shape meaning.
- Integration of knowledge and ideas
- A PERT Reading skill area: evaluating arguments and comparing information across sources.
- Explicit detail question
- Asks for information stated directly in the passage; locate the exact line.
- Vocabulary acquisition
- A PERT Reading skill: determining the meaning of words and phrases, including academic vocabulary.
- Analyzing word choice
- Examining how an author's specific words shape meaning, tone, and impact.
- Central idea vs. theme
- A central idea is the main point of informational text; a theme is the underlying message of literary text.
- Objective summary
- A summary that captures the main points without inserting the reader's own opinions.
- Logical reasoning in text
- Tracing how an author builds a claim with premises and evidence to reach a conclusion.
- Author's perspective
- The viewpoint or attitude that shapes how an author presents a topic.
- Textual evidence
- Specific words, phrases, or lines from a passage that support an answer or inference.
- Comparing genres
- Recognizing how a poem, story, editorial, or report each handles a topic differently.
- Implicit message
- An idea a text suggests through detail and tone rather than stating directly.
- Determining importance
- Distinguishing the central points of a passage from minor supporting details.
- Reading critically
- Questioning an author's claims, evidence, assumptions, and possible bias while you read.
- Figurative vs. literal meaning
- Literal meaning is exact; figurative meaning uses devices like metaphor to imply more.
- Connotative meaning
- The feelings or associations a word evokes beyond its dictionary definition.
- Cause-effect signal words
- Words like 'because', 'therefore', 'as a result', and 'since' that signal cause-and-effect relationships.
- Sequence signal words
- Words like 'first', 'next', 'then', and 'finally' that signal the order of events.
- Contrast signal words
- Words like 'however', 'but', 'on the other hand', and 'whereas' that signal a difference.
- Evaluating an argument
- Judging whether a claim is supported by sufficient, relevant, and credible evidence.
- Reading for college readiness
- PERT Reading checks whether you can analyze college-level texts without remediation.
- Annotating a passage
- Marking key ideas, transitions, and evidence as you read to answer questions faster.
- Distinguishing supported vs. unsupported
- Choosing the inference the passage's evidence forces, not one merely possible.
- Subject-verb agreement
- A singular subject takes a singular verb and a plural subject takes a plural verb (the dog runs / the dogs run).
- Pronoun-antecedent agreement
- A pronoun must match the noun it replaces in number and gender (each student … his or her work).
- Pronoun case
- Use subjective pronouns (I, he, she, they) as subjects and objective pronouns (me, him, her, them) as objects.
- Verb tense
- The form of a verb that shows time — past, present, or future; keep tense consistent unless the time changes.
- Verb tense shift
- An error in which a sentence wrongly switches tenses (e.g. past to present); fix it for consistency.
- Sentence fragment
- An incomplete sentence missing a subject, a verb, or a complete thought.
- Run-on sentence
- Two independent clauses joined incorrectly without proper punctuation or a conjunction.
- Comma splice
- A run-on error in which two independent clauses are joined by only a comma.
- Independent clause
- A group of words with a subject and verb that can stand alone as a sentence.
- Dependent clause
- A clause with a subject and verb that cannot stand alone (begins with words like because, although, when).
- Coordinating conjunction
- A word that joins equal parts — for, and, nor, but, or, yet, so (FANBOYS).
- Subordinating conjunction
- A word that begins a dependent clause (because, although, since, while, if).
- Parallel structure
- Using the same grammatical form for items in a list, pair, or comparison (reading, writing, and editing).
- Modifier
- A word or phrase that describes another word in a sentence (adjective, adverb, or phrase).
- Misplaced modifier
- A modifier placed too far from the word it describes, causing confusion; move it next to that word.
- Dangling modifier
- A modifier with no clear word to describe in the sentence; fix it by adding the missing subject.
- Comma rules
- Use commas between items in a series, after introductory elements, and before a conjunction joining two clauses.
- Semicolon
- Joins two closely related independent clauses, or separates list items that already contain commas.
- Colon
- Introduces a list, explanation, or quotation after a complete independent clause.
- Apostrophe
- Shows possession (the dog's bone) or marks a contraction (it's = it is); not used for simple plurals.
- Its vs. it's
- 'Its' is possessive (the cat licked its paw); 'it's' is a contraction of 'it is' or 'it has'.
- Their / there / they're
- 'Their' shows possession, 'there' indicates place, 'they're' means 'they are'.
- Your vs. you're
- 'Your' shows possession (your book); 'you're' is a contraction of 'you are'.
- To / too / two
- 'To' is a preposition or infinitive marker, 'too' means also or excessively, 'two' is the number 2.
- Capitalization
- Capitalize the first word of a sentence, proper nouns, and the pronoun 'I'.
- Quotation marks
- Enclose direct quotations and certain titles; periods and commas go inside them in U.S. usage.
- Active voice
- The subject performs the action (The committee approved the plan) — usually clearer and more direct.
- Passive voice
- The subject receives the action (The plan was approved by the committee) — wordier; prefer active voice.
- Topic sentence (writing)
- A sentence that states a paragraph's main idea, usually placed at the beginning.
- Transition word
- A word or phrase that signals the relationship between ideas (however, therefore, meanwhile, furthermore).
- Organization of ideas
- Arranging a text logically with an effective opening, clear transitions, and a strong closing.
- Thesis statement
- A single sentence that states the central claim or position of an essay.
- Wordiness
- Using more words than needed; concise, precise writing is preferred on the PERT Writing subtest.
- Redundancy
- Needlessly repeating the same idea (e.g. 'free gift', 'past history'); eliminate the repeated word.
- Concise writing
- Expressing ideas clearly in the fewest necessary words; the clearest, shortest correct option usually wins.
- Word choice (diction)
- Selecting precise, appropriate words that fit the sentence's meaning and tone.
- Sentence variety
- Mixing short and long, simple and complex sentences to keep writing engaging.
- Combining sentences
- Joining short, choppy sentences with conjunctions or clauses to improve flow.
- Conventions of English
- The standard rules of grammar, usage, punctuation, capitalization, and spelling.
- Pronoun shift
- An error that wrongly switches person (e.g. 'one … you'); keep the point of view consistent.
- Double negative
- Using two negatives in one clause (e.g. 'don't have no'); standard English uses only one.
- Comparative form
- Used to compare two things, usually with '-er' or 'more' (taller, more careful).
- Superlative form
- Used to compare three or more things, usually with '-est' or 'most' (tallest, most careful).
- Good vs. well
- 'Good' is an adjective (a good meal); 'well' is an adverb (she sings well).
- Fewer vs. less
- Use 'fewer' for countable items (fewer cars); use 'less' for uncountable amounts (less water).
- Subject of a sentence
- The noun or pronoun that performs the action or is described.
- Predicate
- The part of a sentence containing the verb that tells what the subject does or is.
- Antecedent
- The noun a pronoun refers back to.
- Idiomatic usage
- Standard, conventional phrasing (e.g. 'interested in', not 'interested on').
- Relevance
- Whether a sentence supports the paragraph's point; cut sentences that do not.
- Sentence order
- Arranging sentences so ideas build logically within a paragraph.
- Conjunctive adverb
- A word like 'however', 'therefore', or 'moreover' that links clauses; often follows a semicolon.
- Effective opening
- An introduction or topic sentence that clearly previews what the paragraph or essay is about.
- Effective conclusion
- A closing that restates the main point and gives the reader a sense of completion.
- Standard written English
- The formal, edited English the PERT Writing subtest expects — correct grammar, usage, and mechanics.
- Establishing a thesis
- Stating a clear central claim or position early so the rest of the writing supports it — a PERT Writing skill.
- Sustaining focus
- Keeping every sentence on the chosen topic or argument without drifting — a PERT Writing skill.
- Supporting an argument
- Backing a claim with reasons, examples, and evidence rather than mere assertion.
- Synthesizing sources
- Combining information from multiple texts into a coherent, well-supported point.
- Citing data and opinions
- Accurately representing and attributing the data, conclusions, and views of others.
- Acknowledging counterarguments
- Recognizing competing arguments or information while still establishing your own claim.
- Conventions, usage, mechanics
- The three layers PERT Writing tests: grammar conventions, correct usage, and mechanics (punctuation, spelling, capitalization).
- Commonly confused words
- Word pairs the PERT tests, such as affect/effect, then/than, accept/except, and lie/lay.
- Affect vs. effect
- 'Affect' is usually a verb meaning to influence; 'effect' is usually a noun meaning a result.
- Then vs. than
- 'Then' refers to time or sequence; 'than' is used in comparisons.
- Accept vs. except
- 'Accept' means to receive; 'except' means to leave out or exclude.
- Coordination
- Joining equal ideas with a coordinating conjunction (and, but, or) — a PERT sentence-structure skill.
- Subordination
- Making one idea dependent on another using a subordinating conjunction (because, although) to show emphasis.
- Avoiding inappropriate shifts
- Keeping verb tense and pronouns consistent within a passage, a tested PERT grammar skill.
- Proper case forms
- Using the correct subjective, objective, or possessive form of a pronoun for its role in the sentence.
- Adjective vs. adverb
- Adjectives modify nouns; adverbs (often ending in -ly) modify verbs, adjectives, or other adverbs.
- Transitional devices
- Words and phrases that connect ideas and signal logical relationships within a passage.
- Style and tone
- The voice and attitude of a piece; effective writing maintains an appropriate, consistent style and tone.
- Conveying information clearly
- Presenting complex information so it is organized, coherent, and easy to follow.
- Revising for clarity
- Editing a draft to remove confusion, wordiness, and errors so the meaning is clear.
- Editing in context
- On the PERT, you choose revisions that improve a draft passage, not isolated grammar trivia.
- Spelling conventions
- Using standard spelling; the PERT Writing subtest tests recognition of correct spelling in context.
- Maintaining agreement
- Keeping subject-verb and pronoun-antecedent agreement throughout a passage.
- Logical paragraph order
- Sequencing sentences and ideas so the passage develops in a logical, connected way.
