CogAT Grades 9-12 cheat sheet

Analogy bridge test: state the relationship as a full sentence linking the first pair (e.g., "A hammer is used to drive a nail"); apply the identical sentence to the answer choices and eliminate any pair that fails it.

Common analogy relationships to name explicitly: synonym, antonym, part-to-whole, whole-to-part, cause-to-effect, function/tool-to-user, category-to-member, degree/intensity, sequence, and object-to-material.

Order rule: in A : B :: C : ?, the direction must match. If A is the category and B is a member, the answer must be a member of C's category, not a category containing C.

Classification (odd-one-out) method: propose a single shared property for four of the five words, then confirm the leftover word lacks that property; a valid answer requires that exactly one word fails the rule.

Watch for the trap answer that shares a topic but not the relationship: two words can both be nautical or both be musical yet have no consistent logical link to the stem pair.

Sentence completion: cover the blank(s), predict a word from the sentence's own logic, then match by meaning; test the chosen word by rereading the whole sentence for grammar and tone.

Signal words flip the blank's polarity: "although, despite, rather than" signal contrast; "because, therefore, since, moreover" signal continuation or cause.

Double-blank items must satisfy both blanks simultaneously; an option that fits one blank but not the other is wrong regardless of how natural the first blank sounds.

Series first pass: compute consecutive differences. A constant difference means linear growth; differences that themselves change by a constant amount mean a second-order (quadratic) pattern.

If differences fail, compute ratios. A constant ratio means geometric growth; a ratio that itself doubles or halves signals an exponential pattern.

Interleaved series rule: when differences alternate in sign or magnitude, split into odd-position and even-position subsequences and solve each separately.

Alternating-operation patterns (×2, +3, ×2, +3 …) and two-step patterns (add then multiply) are common; test the operation pair on the first three terms before applying it forward.

Equation balance rule: whatever operation is applied to one side must be applied to the other; undo addition/subtraction before multiplication/division, and clear parentheses first.

Two-variable word problems: translate each sentence into an equation, then substitute from the equation with the isolated variable; check the solution against the original wording, not just the algebra.

Ratio and proportion: set up as a fraction equality and cross-multiply; for part-to-part ratios, convert to part-to-whole by summing the parts before computing a total.

Average (mean) = sum ÷ count, so sum = mean × count; if a value is added or removed, recompute from the total rather than averaging the averages.

Percentage change = (new − old) ÷ old × 100; successive percentage changes multiply rather than add (a 20% rise then a 20% fall yields a net 4% loss).

Matrix strategy: read across each row to find the transformation, then verify it down each column; the correct answer must satisfy both directions.

Track attributes separately: shape identity, size, rotation angle, fill (solid/outline/striped), position, and count. Most items change one or two attributes at a time.

Rotation versus reflection: a 90° turn moves an asymmetric mark to a predictable side, while a mirror image reverses it without changing orientation; test with a single distinctive feature such as a dot or notch.

Sequence rule: count the elements, note the direction of movement (clockwise/counterclockwise), and measure the step size; the next figure continues the established step.

Additive and subtractive rules: in some puzzles the third figure is the overlap of the first two (shared parts kept) or the difference (shared parts removed).

Classification (odd-one-out) figures: find the property shared by four figures - symmetry, number of sides, closed vs open, number of enclosed regions - and select the figure that violates it.

Paper-folding reasoning: unfold in reverse order, mirroring each hole across the fold line at each step; the number of holes multiplies with each fold that the hole crosses.

Counting invariants: number of sides, number of enclosed regions, number of intersections, and lines of symmetry are the most frequently tested numeric properties in figure sets.

Context clue types: definition (often set off by commas or dashes), restatement ("that is"), contrast ("but, unlike"), and example ("such as"); locate the clue before choosing a meaning.

Prefix families: bene-/mal-, pro-/anti-, pre-/post-, hyper-/hypo-, circum-, and intra-/inter- reliably signal direction, timing, or evaluation.

Suffix families: -ology (study of), -phobia (fear of), -cracy (rule by), -cede/-ceed (go, yield), and -vert (turn) narrow a word's meaning even when the root is unfamiliar.

Connotation check: two options may be denotationally close but differ in charge (thrifty vs. stingy); the surrounding sentence's tone selects between them.

Word pairs with reversed direction (give/take, import/export, ascend/descend) appear as antonym distractors; confirm which direction the stem sentence requires.

If two answer choices both seem to fit, one is usually too broad or too narrow; the correct choice matches the sentence's scope exactly.

Number-puzzle grids: test each row, column, and diagonal for a constant sum, product, or difference before assuming a pattern; the rule must hold for every complete line.

Missing-number puzzles with a center value: check whether the center equals the sum, product, or average of the surrounding numbers, and whether an operation is applied before or after combining.

Shortcut for comparing fractions: cross-multiply (a/b vs c/d compares ad vs cb) instead of converting to decimals.

Divisibility shortcuts: by 3 if digits sum to a multiple of 3; by 9 if digits sum to a multiple of 9; by 4 if the last two digits form a multiple of 4; by 6 if divisible by both 2 and 3.

Estimation guard: bound the answer before computing (e.g., 48 × 21 lies between 48 × 20 and 48 × 22), which eliminates most distractors without full arithmetic.

Work-rate and distance setups: distance = rate × time; combined work rate adds the individual rates (1/a + 1/b = 1/t).

Net folding: opposite faces of a cube net never share an edge; identify opposite pairs first, then eliminate answers showing two opposite faces adjacent.

Cube rotation: a cube has 24 distinct orientations; a single visible face constrains the hidden faces only through the faces adjacent to it.

Fold-line symmetry: a folded shape's crease becomes a line of symmetry when unfolded, so hole patterns are mirror images across each crease.

Transparent vs. opaque folding: for transparent paper the hole is punched through all layers; for opaque stacks only the top layer is affected unless stated otherwise.

Rotation preserves shape and size; scaling changes size only; shearing or stretching changes angles - check angles when a transformation's type is ambiguous.

Embedded-figure items: scan for the target's distinctive corner or angle first, since that feature is hardest to disguise inside a larger drawing.

Inference must follow necessarily from the stated text; an answer requiring one extra assumption beyond the passage is out of scope.

Scope words decide correctness: "all, never, only, must" are strong claims needing strong textual support, while "some, may, often" are easier to support.

Main-idea questions ask what the passage is mostly about, not what its first sentence says; the correct choice covers the whole passage rather than one detail.

Author's tone is signaled by evaluative adjectives and adverbs; neutral reporting and advocacy are distinguishable by whether the author endorses or merely describes.

Cause-and-effect in a passage is marked by "because, led to, resulted in, due to"; a mere correlation in the text cannot be restated as causation.

Set problems: total = group A + group B − both + neither; solve for the unknown term rather than counting members individually.

Venn reasoning with three groups: add the three pairwise overlaps, subtract the triple overlap once, and reconcile against the stated total.

Ordering and ranking puzzles: draw a single line and place entities from the most constrained clue first; a clue fixing a position relative to both ends is the strongest starting point.

If-then logic: "if P then Q" is equivalent to "if not Q then not P" (contrapositive); the reversed form "if Q then P" is not valid.

Counting arrangements: for n distinct items in a row there are n! orders; if some items are identical, divide by the factorial of each repeated group.

Unit and label checks: verify whether an answer is asked for as a rate, a total, or a difference before selecting among numerically similar choices.

Alternating rules: some series apply rule A on odd steps and rule B on even steps; test the pattern across two steps before predicting the next figure.

Progressive change: rotation angle, size, or element count may increase by a constant increment (e.g., +45°, +90°, +135°), so measure the increment rather than assuming a fixed step.

Element migration: a dot or mark may move one position clockwise per step while another moves counterclockwise; track each element independently.

Superposition with cancellation: when two figures combine, overlapping lines may cancel (XOR) rather than merge; compare where lines disappear as well as where they appear.

Color/fill inversion: a solid-to-outline flip may alternate every step or occur once at a fixed point; count the steps between changes to establish the period.

Sequence of squares and cubes: 1, 4, 9, 16, 25 (differences 3, 5, 7, 9) and 1, 8, 27, 64 (differences 7, 19, 37) are frequent; recognize them by their difference signatures.

Fibonacci-type rules: each term is the sum of the two preceding terms; test by adding the first two terms and comparing with the third.

Difference-of-differences method: for any polynomial sequence, repeated differencing eventually yields a constant row; the number of differencing rounds equals the polynomial's degree.

Arithmetic sequence: nth term = first term + (n − 1) × common difference; sum = n × (first + last) ÷ 2.

Geometric sequence: nth term = first term × ratio^(n − 1); a ratio between 0 and 1 produces a decreasing sequence that never reaches zero.

Substitution check: after solving for a variable, plug it back into the original equation, not a rearranged one, to catch sign errors.

Analogy figures (A is to B as C is to ?): describe the change from A to B in words (e.g., "rotated 90° clockwise and halved"), then apply exactly that change to C.

Distractor design: wrong answers typically apply the right transformation to the wrong element or the wrong transformation to the right element; verify both.

Symmetry classification: count lines of symmetry and check rotational symmetry order; figures sharing the same symmetry profile belong together.

Region counting: the number of enclosed regions (not lines) often defines the class; count regions before comparing shapes.

Open vs. closed figures and straight vs. curved edges are reliable binary classifiers when shape identity varies across the set.

Degree relationships (warm : hot :: cool : cold) require matching intensity order; a pair with the same direction but wrong magnitude is a distractor.

Function relationships pair an object with its purpose (scalpel : surgeon :: loom : weaver); the answer must match both the tool and the agent role.

Whole-to-part chains can be nested (tree : branch :: branch : twig); confirm the level of the hierarchy matches the stem.

Antonym pairs are often disguised by shared topic; state the relationship as "X means the opposite of Y" and test each option against that sentence.

Precision rule: choose the option whose relationship is most specific; "animal : dog" is weaker than "dog : puppy" when the stem encodes age or stage.

Average speed for a round trip with equal distances = 2ab ÷ (a + b), the harmonic mean, not the arithmetic mean of the two speeds.

Mixture problems: amount of substance = concentration × volume; when two solutions combine, total substance and total volume each add.

Ratio scaling: if a : b = 3 : 5 and the total is 40, one part equals 40 ÷ 8 = 5, so the parts are 15 and 25.

Percent of a number: convert to a decimal multiplier (35% → 0.35); "of" means multiply, and "what percent" means solve for the multiplier.

Weighted average = (sum of value × weight) ÷ (sum of weights); a simple average is valid only when all weights are equal.

Rotation direction check: a clockwise turn of the figure moves a top-edge mark to the right edge; a counterclockwise turn moves it to the left edge.

Mirror versus rotate test: rotating a letter-like figure preserves its handedness, while mirroring reverses it; use an asymmetric feature such as a flag or tail.

Unfolding order matters: reverse the folds from last to first, reflecting the hole across each crease, to reconstruct the flat pattern.

Hole count rule: each fold doubles the number of layers at the punched location, so a punch through all layers yields 2^k holes for k folds crossed.

Stack-and-cut items: identical shapes stacked and cut produce identical cutouts in every layer; a rotated layer produces a rotated cutout.

Even/odd rules: even × anything = even; odd × odd = odd; even + odd = odd; odd + odd = even.

Prime recognition up to 100: test divisibility by 2, 3, 5, and 7 only; if none divides, the number is prime.

Consecutive integers n, n+1, n+2 sum to 3n + 3, so the sum is always divisible by 3.

Place value: in a two-digit number 10a + b, reversing the digits gives 10b + a, and the difference is 9(b − a), always a multiple of 9.

Remainder arithmetic: a number leaving remainder r when divided by d has the form kd + r; test candidate answers by this form rather than long division.

Attribute independence: in a 3×3 matrix, one attribute may change across rows while another changes down columns; solve each attribute separately and combine.

Constant-in-row rule: if every figure in a row shares a feature (same fill, same number of sides), that feature is fixed and the answer must carry it.

Progression across rows: element counts may increase by one per column (1, 2, 3) and per row; check both directions for consistency.

Intersection rule: where two figures overlap, the result may keep only the shared region, only the non-shared region, or the union; determine which by inspecting the completed cells.

Elimination by single attribute: reject any option violating an attribute already fixed by the row or column, then compare survivors on the remaining attribute.

Category level matters: four words may all be vehicles while one is a bicycle, but if all five are vehicles the rule must be finer, such as "motorized" or "four-wheeled."

Part-of-speech consistency: if four words are nouns and one is a verb, the grammatical outlier is a candidate only if no semantic rule fits four others.

Abstract vs. concrete: a set of concrete objects with one abstract concept usually classifies on that axis when no other rule holds.

Animate vs. inanimate and natural vs. man-made are the two most common binary splits in classification items.

Verify uniqueness: a correct classification answer must be the only word that fails the shared rule; if two words fail, the rule is wrong.

Area formulas: rectangle lw, triangle ½bh, circle πr², trapezoid ½(b₁ + b₂)h; perimeter of a circle is 2πr.

Pythagorean triples to recognize: 3-4-5, 5-12-13, 8-15-17, and their multiples; a right triangle with two known legs can be solved without a calculator.

Angle facts: angles on a straight line sum to 180°, around a point to 360°, and vertical angles are equal; parallel lines cut by a transversal create equal alternate interior angles.

Volume: rectangular prism lwh, cylinder πr²h; surface area of a cube is 6s².

Scale factors: if lengths scale by k, areas scale by k² and volumes by k³.

Count before comparing: tally sides, vertices, enclosed regions, and intersections for each figure; the discriminating attribute is usually a count that differs by exactly one.

Line-type comparison: straight versus curved, parallel versus intersecting, and open versus closed are the fastest binary filters.

Shading patterns: solid, hatched, and outline fills may cycle in a fixed order (solid → hatched → outline); identify the cycle length before predicting.

Position rules: elements may be anchored to corners, edges, or the center; check whether the anchor point stays fixed while the element's shape changes.

Size progression: diameters or side lengths may increase arithmetically; measure the increment across the first three figures to predict the fourth.