Short lessons and technique guides
Skill lessons give you the rule, the trap to avoid and a worked example. Technique lessons (backsolving, passage mapping, transition logic and more) show how to attack a question type, with worked examples from real questions. When you miss a question in a drill, the matching lesson opens right under the explanation.
English
Technique lessons
How to attack a question type, with worked examples from real questions.
- ConcisionPick the shortest choice that still says everything the sentence needs, and nothing it already said.
- Transition logicName the relationship between the sentence before and the sentence after (addition, contrast, cause, example, sequence), then pick the word that says it.
- Punctuation edge casesThree marks cause the tricky errors: the colon, the semicolon and paired commas or dashes. Each has one test.
Conventions of Standard English
- PunctuationCommas, semicolons, colons, dashes and apostrophes each have one job. Match the mark to the job.
- Subject-Verb AgreementA verb must match its subject in number. Find the true subject first, then check the verb.
- Sentence BoundariesEvery sentence needs a subject and a verb and must express a complete thought. Two complete sentences cannot be joined by only a comma.
- Parallel StructureItems in a list or comparison must be in the same grammatical form.
- Verb Forms and TensesKeep tense consistent unless the time really changes, and use the right form after helping verbs.
- Pronouns and PossessivesA pronoun must match its noun in number and be clear about who or what it means. Apostrophes show possession or a contraction, never a plural.
Knowledge of Language
- Word ChoicePick the word with the exact meaning and the right tone, not the one that merely sounds impressive.
- ConcisenessWhen two choices say the same thing, choose the shorter one, as long as no meaning is lost.
- Modifier PlacementA modifier describes the word closest to it. Put it next to the word it describes.
- Style and ToneMatch the style of the passage. Keep the same level of formality and voice throughout.
Production of Writing
- TransitionsA transition tells the reader how two ideas relate. Decide the relationship first, then pick the word.
- OrganizationOrder ideas so each sentence builds on the last. Look for clues that link sentences.
- Supporting DetailsA good supporting detail is specific, relevant, and does the job the question asks for.
- Introductions and ConclusionsAn introduction sets up the topic. A conclusion wraps up the main idea without bringing in a new one.
- Relevance and PurposeDecide whether a sentence belongs by asking what the paragraph is for.
Reading
Technique lessons
How to attack a question type, with worked examples from real questions.
- Passage mappingSpend 60–90 seconds writing a one-line map of each paragraph, then use the map to jump straight to the answer.
- Evidence matchingProve every answer with a line in the passage. If you cannot point to the words, the choice is probably an outside idea.
- Paired-passage strategySummarize each author’s position separately first, then compare. Questions about both passages are just those two summaries side by side.
- Timing per passageBudget about ten minutes a passage: one for the map, the rest for the questions, with the hardest big-picture questions last.
Key Ideas and Details
- Main idea and summaryThe main idea is the one point the whole passage supports. It is bigger than any single paragraph.
- Finding detailsDetail questions are answered directly in the text. Find the lines, then match the meaning.
- Making inferencesAn inference is the one conclusion the text strongly supports, though it never says it outright.
- Fiction: character and motiveIn literary narratives, read the behavior and the small details for what the character feels.
- Pacing and strategy for the sectionYou have about 10 minutes per passage. Read for the big picture, then go back for details.
Craft and Structure
- Words in contextUse the sentence, not your first definition, to decide what a word means here.
- Author's purpose and structureAsk why the author included something and how the passage is organized.
- Tone and attitudeTone comes from word choice. Look at the adjectives, verbs and details the author picks.
Integration of Knowledge and Ideas
Math
Technique lessons
How to attack a question type, with worked examples from real questions.
- BacksolvingWhen the choices are numbers, test them in the problem instead of solving for the unknown.
- Picking numbersReplace variables or unknown amounts with easy numbers and compare the choices, instead of proving something in general.
- Desmos graphs and tablesUse the built-in calculator to read answers off a graph or a table when the algebra is long.
- Units and ratesCarry the units through the calculation so they cancel into the unit the question asks for.
- Probability trapsThree setups cause most lost points: “at least one”, drawing without replacement, and “given that”.
Number & Quantity
- Arithmetic & FractionsGet a common denominator to add or subtract, multiply straight across, and flip to divide.
- Complex NumbersTreat i like a variable, but replace i² with −1.
- Exponents & RadicalsUse the exponent rules and write radicals as fractional powers.
- Number PropertiesKnow primes, factors, multiples and the rules for odd and even numbers.
- PercentsPercent means per hundred. Convert to a decimal and multiply, and use the original as the base for change.
- Ratios & ProportionsSet two equal ratios and cross-multiply.
- Units, Rates & Scientific NotationCarry units through the problem, and let them cancel.
Algebra
- Linear EquationsUndo operations in reverse order and keep the equation balanced.
- Linear InequalitiesSolve like an equation, but flip the sign when you multiply or divide by a negative.
- Systems of EquationsUse substitution or elimination to find where two lines meet.
- Quadratic EquationsFactor, complete the square or use the quadratic formula.
- Polynomials & FactoringLook for a common factor, then a pattern.
- Radical & Absolute-Value EquationsIsolate the radical or the absolute value, then undo it, and always check.
- Rational ExpressionsFactor, cancel common factors, and watch for values that make a denominator zero.
- Rearranging FormulasTreat the variable you want as the unknown and undo the operations around it.
- Slope & Graphs of LinesSlope is rise over run. Use y = mx + b to read a line.
- Linear Modeling Word ProblemsIdentify the starting amount and the rate of change.
- Exponential Growth & DecayA quantity that changes by a constant percent follows y = a(1 ± r)ᵗ.
- Sequences & SeriesArithmetic sequences add a constant. Geometric sequences multiply by a constant.
Integrating Essential Skills
- Systems of InequalitiesGraph each inequality and find the overlap.
- Work, Mixture & DistanceUse rate × time = amount, and add rates when people work together.
- Multi-Step Word ProblemsBreak the story into small steps and label each quantity.
- Mixed ACT MathMixed sets test whether you can recognize the topic first.
Functions
- Function Notationf(x) is the output for input x. Substitute the input everywhere x appears.
- Composition & InversesWork from the inside out for composition, and swap x and y for inverses.
- Domain, Range & BehaviorDomain is the allowed inputs. Range is the possible outputs.
- Graph TransformationsChanges inside the parentheses move the graph sideways, outside move it up or down.
- LogarithmsA logarithm is an exponent: log_b(x) = y means b^y = x.
- Polynomial Functions & ZerosZeros are where the graph crosses the x-axis. Factor to find them.
- Quadratic Functions & ParabolasThe vertex form y = a(x − h)² + k shows the vertex (h, k).
Geometry
- Angles & Parallel LinesKnow which angles are equal and which add to 180°.
- CirclesKnow circumference, area, arcs and sectors.
- Composite Area & PerimeterSplit the shape into rectangles, triangles and circles, and add or subtract areas.
- Coordinate GeometryUse the distance and midpoint formulas.
- Law of Sines & CosinesUse these for triangles that are not right triangles.
- Polygons & Angle SumsInterior angles of an n-sided polygon add to (n − 2) · 180°.
- Right-Triangle TrigonometrySOH-CAH-TOA relates the angles to the sides.
- Similar & Congruent FiguresSimilar figures have equal angles and proportional sides.
- Triangles & the Pythagorean TheoremIn a right triangle, a² + b² = c², where c is the hypotenuse.
- Trig Graphs & IdentitiesSine and cosine waves have amplitude, period and shifts.
- Unit Circle & RadiansRadians measure an angle by arc length. 180° = π radians.
- Volume & Surface AreaPick the formula that matches the solid.
Statistics & Probability
- Mean, Median & ModeMean is the average, median is the middle, mode is the most common.
- Probability BasicsProbability = favorable outcomes / total outcomes.
- Compound & Conditional ProbabilityMultiply for 'and' (adjusting for no replacement), and add for 'or'.
- CountingUse the multiplication rule, permutations and combinations.
- Reading Tables & GraphsRead the labels first, then find the values.
- Sampling & Margin of ErrorA sample can estimate a population. The margin of error shows the likely range.
- Scatterplots & Lines of Best FitA line of best fit summarizes a trend. Use its equation to predict.
- Spread & Standard DeviationStandard deviation measures how far the data are from the mean.