Step-by-step guides for every tested skill — worked examples, key formulas, and practice.
106 concepts
Master one-variable linear equations for the SAT: the balance principle, variables on both sides, fractions, and the parameter questions that hide "no solution" and "infinitely many."
Read slope as a rate of change, move fluently between slope-intercept, standard, and point-slope form, and handle the parallel and perpendicular slope relationships the SAT rewards.
Graph SAT lines from intercepts or from slope, recognize horizontal and vertical lines, and use parallel and perpendicular slopes to analyze lines in the xy-plane.
Solve SAT systems of two linear equations with substitution and elimination, and tell how many solutions a system has.
Solve and graph SAT linear inequalities in one and two variables, including when to flip the sign and how to shade the correct region.
Solve SAT absolute value equations and inequalities by treating absolute value as distance and splitting into two cases.
Solve SAT quadratic equations with factoring, the square-root method, completing the square, and the quadratic formula — and use the discriminant to count real solutions.
Factor SAT polynomials using GCF, difference of squares, trinomials, and grouping — then use the factored form to solve equations, find zeros, and simplify expressions.
Add, subtract, and multiply SAT polynomials, combine like terms carefully, and use the factor and remainder theorems to test divisibility.
Master SAT function notation: evaluate f(x), read function values from graphs and tables, compose functions, solve for an input, and find domain and range.
Understand SAT exponential functions y = a·b^x: interpret growth and decay, convert percent rates to factors, read a and b, and separate exponential from linear models.
Apply SAT exponent laws with negative and fractional powers, convert radicals to rational exponents, and simplify roots and expressions step by step.
Solve SAT systems with a line and a parabola by substitution, and use the discriminant to count how many times two curves intersect.
Set up and solve SAT proportions, scale quantities up or down, split totals by a ratio, and recognize proportional relationships in tables, graphs, and equations.
Compute percent of a number, percent increase and decrease, successive (chained) percent changes, and reverse percent problems for SAT Math data questions.
Work with unit rates, speed, density, and combined-work problems, and carry out multi-step dimensional-analysis conversions on SAT Math.
Read SAT scatterplots, identify positive, negative, or no association, and use a line of best fit to predict values, interpret slope and intercept, and reason about residuals.
Compute mean, median, and mode, reason about range and standard deviation, and judge the effect of outliers on SAT statistics questions.
Compute simple and compound probability and read conditional probability from two-way tables on SAT Problem-Solving and Data Analysis questions.
Understand random sampling, generalizing to a population, margin of error, and confidence-interval reasoning on the SAT.
Use the Pythagorean theorem to find missing sides of right triangles, measure distances, and solve 2D and 3D SAT geometry problems.
Master SAT area, perimeter, surface area, and volume for polygons, circles, prisms, cylinders, cones, and spheres — including composite figures and scaling.
Learn SAT angle relationships, parallel lines cut by a transversal, the triangle angle sum, exterior angles, and similar and congruent triangles.
Solve SAT circle problems: circumference, area, arc length, sector area, central vs inscribed angles, and the circle equation.
Use SOH-CAH-TOA to find sides and angles of SAT right triangles, plus the sine-cosine complementary-angle relationship.
Use the 45-45-90 and 30-60-90 SAT special right triangle ratios to find side lengths instantly without a calculator.
Solve, model, and graph linear equations and inequalities for the ACT Math section, including slope, intercepts, and word-problem setups.
Solve ACT systems of equations with substitution and elimination, interpret how many solutions exist, and set up two-variable word problems.
Factor, solve, and graph quadratics for the ACT Math section, including the quadratic formula, the vertex, and the discriminant.
Master exponent rules, radicals, rational and negative exponents, and simplifying roots for the ACT Math section.
Understand logarithms as inverse exponents, convert between log and exponential form, and apply the log rules on ACT Math.
Evaluate functions, read function notation, and shift, stretch, and reflect graphs for the ACT Math section.
Master arithmetic and geometric sequences and their sums for the ACT Math section, including nth-term formulas and finite-series shortcuts.
Learn matrix addition, subtraction, scalar and matrix multiplication, and determinants — ACT-favorite operations tested on the Math section.
Understand the imaginary unit i, powers of i, and complex-number arithmetic and conjugates for the ACT Math section.
Master ACT coordinate geometry: slope, distance, midpoint, and parallel and perpendicular lines in the xy-plane, plus how to read them off a graph.
Read and write ACT circle and ellipse equations: find the center and radius, complete the square, and identify a conic from its equation.
ACT triangle essentials: angle-sum and exterior-angle rules, similar and congruent triangles, Pythagorean triples, and the special right triangles.
Master right-triangle trigonometry for the ACT Math section: the sine, cosine, and tangent ratios, inverse trig for angles, and cofunctions.
Solve oblique (non-right) triangles on the ACT Math section with the law of sines and the law of cosines, and choose the right tool for each setup.
Compute perimeter, area, surface area, and volume of standard 2D and 3D figures for the ACT Math section — including circles, prisms, cylinders, and cones.
Compute simple and compound probability for the ACT Math section, plus factorials, permutations, and combinations for counting outcomes.
Master the ACT Math measures of center and spread — mean, median, mode, range, and weighted mean — plus missing-value and combined-group setups.
Solve ACT Math percent, percent-change, ratio, proportion, and unit-rate problems, including discounts, markups, and reverse-percent setups.
A repeatable method for finding the central idea of an SAT Reading passage and telling it apart from the supporting details — including how to beat "too narrow," "too broad," and "half-right" trap answers.
Master both flavors of SAT Command of Evidence — textual and quantitative — by pinning the exact claim first, then choosing the quotation, finding, or data point that proves that precise claim and nothing else.
The SAT method for drawing the one conclusion a passage forces — a small, safe step from the stated facts — and for eliminating the overreaching "smart-sounding" answers that make inference questions hard.
Master the SAT’s vocabulary-in-context questions with a predict-then-match method that handles second meanings, connotation, and contrast-driven blanks.
Answer SAT structure-and-purpose questions by naming the job a sentence does and mapping a passage’s begin-turn-end arc, not by summarizing content.
Compare two paired SAT passages: state each position, name the precise relationship, and derive how one author would respond to the other.
Choose the verb that matches the real subject of an SAT sentence by ignoring interrupting phrases, handling "or/nor," collective nouns, and inverted word order.
Master SAT transition questions by reading both sentences, naming the logical relationship between them, and matching the connector to it — contrast, cause, example, or continuation.
Learn the SAT rhetorical synthesis method: read the goal first, turn it into a checklist, and pick the choice that accomplishes the exact task — not merely the one that is true.
Fix SAT comma splices, run-ons, and fragments by learning when to use periods, semicolons, colons, commas, and dashes between clauses.
Fix SAT pronoun-antecedent agreement, pronoun case, and dangling or misplaced modifiers by matching words to the nouns they truly refer to.
Choose the right SAT verb tense and form: keep tenses consistent, resolve tricky subject-verb agreement, and match verbs to their true subjects.
Master ACT English comma questions by testing whether the underlined comma marks a real boundary — an interrupter, a list, a joined clause — instead of guessing by pauses.
Learn the ACT English rules for semicolons, colons, and dashes by testing what sits on each side of the mark — a full sentence, a list, or an interruption.
Fix ACT English run-ons, comma splices, and fragments by counting independent clauses and choosing punctuation that legally joins or separates them.
Master ACT English agreement questions by finding the true subject and antecedent, ignoring the words the test buries in between, and matching number every time.
Choose the right ACT English verb by reading the sentence for time cues, keeping tense consistent, and using standard past and participle forms.
Fix ACT English dangling modifiers and faulty parallelism by putting descriptive phrases next to what they describe and matching items in a series.
Master ACT English concision questions by cutting repeated ideas and empty filler — when several choices are all correct, the shortest clear one almost always wins.
Learn ACT English word-choice questions by matching diction to context — precise meaning, idiom, register, and consistent tone — not by which word "sounds smartest."
Master ACT English transition questions by reading the two ideas a connector joins — contrast, cause, addition, or sequence — instead of picking a word that merely sounds smooth.
Master ACT English rhetoric questions — adding, deleting, or keeping a sentence — by judging relevance to the paragraph’s purpose, not by whether the detail is interesting or true.
A repeatable method for naming the main idea of an ACT Reading passage or paragraph — and for beating the "too narrow," "too broad," and "half-right" trap answers that make these questions hard.
How to answer ACT Reading detail questions fast and correctly — locate the exact line, match it literally, and refuse the "true in real life but not in the passage" trap that costs the most points.
The ACT method for drawing the one conclusion a passage supports — a small, safe step beyond the stated facts — and for eliminating the overreaching answers that make inference questions hard.
How to answer ACT Reading vocabulary-in-context questions — using the sentence, not the dictionary, to pick the meaning a word carries in this specific passage.
How to answer ACT Reading questions about why an author wrote something and how the passage is built — purpose, function of a paragraph, and rhetorical choices.
A strategy for the ACT Reading paired-passage set — reading two texts on one topic and answering single-passage, cross-passage, and relationship questions accurately.
How to read values, units, and axes off ACT Science tables and graphs — and how to pull a single fact out of a busy figure without needing any outside science knowledge.
Spot direct and inverse trends in ACT Science data, describe how two variables relate, and interpolate or extrapolate a value beyond the rows the table gives you.
Identify independent, dependent, and controlled variables in an ACT Research Summary, spot the control group, and explain why an experimental step was included — all from the passage.
Judge whether ACT Science data actually supports a stated conclusion, tell a supported claim from an overreach, and name the new evidence that would confirm a result.
Master the ACT Science Conflicting Viewpoints passage: compare two scientists' hypotheses, find where they agree and disagree, and match new evidence to the side it supports.
Test a model or prediction against ACT Science data: decide whether results support it, which data would strengthen or weaken it, and how to refine a model that partly fails.
Reason about integers the way the GRE wants — parity, primes, factors and multiples, divisibility tests, remainders, GCF/LCM, and consecutive-integer facts — so "must be true" questions become quick.
Move fluently among fractions, decimals, and percents, and master the GRE’s favorite traps: percent of a number, percent increase and decrease, successive percent changes, and "percent more/less than."
Set up and solve GRE ratio problems with confidence — part-to-part vs. part-to-whole, combining ratios through a shared term, direct and inverse proportion, unit rates, and combined work and speed.
Master the exponent rules, fractional exponents, radicals, rationalizing, and scientific notation that power a large share of GRE Quant.
Learn the four fixed answer choices, the "cannot be determined" mindset, and the plug-in discipline that turns GRE Quantitative Comparison into free points.
Solve linear equations, two-variable systems, inequalities, and absolute-value statements — and read their solution sets on a number line — for the GRE Quant section.
Simplify and factor expressions, expand with FOIL, and translate GRE age, mixture, interest, and work stories into equations you can actually solve.
Factor, expand, and solve quadratics for GRE Quant — zero-product property, the quadratic formula, the discriminant, and reading a parabola.
Master function notation, slope, distance, midpoint, and the equation of a line for GRE Quant — plus reading values straight off a graph.
Master GRE angle relationships — complementary, supplementary, and vertical angles, the straight-line and full-turn sums, and parallel lines cut by a transversal — then chase unknown angles with confidence.
Command GRE triangle and polygon rules — the angle sum, exterior-angle theorem, triangle inequality, area, the Pythagorean special right triangles, similarity, and the interior-angle sum of any n-gon.
Master GRE circle geometry — radius, diameter, circumference, area, arcs and sectors, inscribed and central angles, tangents, and circles in the coordinate plane.
Master GRE solid geometry — volume and surface area of boxes, cubes, and cylinders, cone and sphere formulas, the space diagonal of a box, and combining 2D facts inside 3D figures.
Master mean, median, mode, range, quartiles, weighted averages, and standard deviation as a spread measure for GRE Quant data-analysis questions.
Read the bell curve, apply the 68-95-99.7 empirical rule, standardize with z-scores, and reason about percentiles for GRE Quant.
Master GRE probability — complements, the addition and multiplication rules, "at least one" via complement, conditional probability, and geometric probability.
Count arrangements and selections on the GRE with the fundamental counting principle, permutations, combinations, and restriction handling.
Read GRE bar graphs, line graphs, tables, and pie charts, then answer multi-step percent and ratio questions from a shared data set.
A repeatable method for filling the single blank in a GRE Text Completion sentence — read the sentence for its logic, predict the word before looking at the choices, and beat the five hard-vocabulary distractors that make these questions so treacherous.
How to solve GRE two- and three-blank Text Completions by treating the blanks as an interdependent system — start where the sentence is most determined, let each filled blank constrain the next, and confirm that the whole set reads as one coherent thought.
The disciplined method for GRE Sentence Equivalence — predict the blank from the sentence, then choose the ONE pair of words that is both contextually right and mutually synonymous, defeating the "fits-but-not-synonyms" and "synonyms-but-wrong-meaning" traps.
A strategy-first guide to the hardest recurring GRE words — using roots and prefixes, connotation, and secondary meanings to reason your way to a word's meaning in context, even when you have never memorized it.
A disciplined method for naming the central point and primary purpose of a dense GRE passage — and for beating the too-narrow, too-broad, and distortion trap answers that make these questions the hardest points to earn on the Verbal section.
How to answer GRE "according to the passage" and NOT/EXCEPT questions with precision — locate the exact stated information, match it literally, and refuse the true-in-general-but-unsupported trap that costs the most points.
How to answer the GRE's hardest reading questions — "the author mentions X in order to…," "the primary function of the second paragraph is…," and "the passage proceeds by…" — by reading a passage as an argument with moving parts rather than a bag of facts.
A disciplined method for answering GRE "the passage implies/suggests" questions — deriving what MUST be true from the text while refusing the seductive answers that reach one inch too far.
How to pin down what a word means AS USED in a GRE passage — using the substitution test and the surrounding logic to defeat the trap that a word’s most familiar meaning is the answer.
A rigorous framework for GRE argument questions — separating conclusion from premise from assumption, then strengthening, weakening, and evaluating with the "which of the following, if true" family.