OAE Mathematics cheat sheet

Number-system hierarchy: naturals ⊂ wholes ⊂ integers ⊂ rationals ⊂ reals ⊂ complex; irrationals sit in the reals but outside the rationals, and every rational has a terminating or eventually repeating decimal expansion.

Divisibility shortcuts: by 3 or 9 when the digit sum is divisible by 3 or 9; by 4 when the last two digits form a multiple of 4; by 8 when the last three digits form a multiple of 8; by 11 when the alternating digit sum is divisible by 11.

Place-value reasoning: multiplying by 10^k shifts the decimal point k places right; dividing shifts it left, so 0.0045 × 10^3 = 4.5 and 62 ÷ 10^4 = 0.0062.

Order of operations: grouping symbols, then exponents and roots, then multiplication and division left to right, then addition and subtraction left to right; a fraction bar acts as an implicit grouping symbol.

Proportional reasoning: a/b = c/d implies ad = bc; unit rate = quantity ÷ units, and scaling both terms of a ratio by the same nonzero factor preserves the ratio.

Percent change = (new − old)/old × 100; successive percent changes multiply rather than add, so a 20% increase followed by a 20% decrease yields 0.96 of the original.

Absolute value |x| is the distance from x to 0 on the number line, so |a − b| gives the distance between a and b; |x| = k has solutions x = k and x = −k for k > 0.

Estimation and reasonableness: round to compatible numbers before computing, then check that the exact result lies within a sensible range; benchmark fractions (1/2, 1/4, 3/4) support quick magnitude comparisons.

Mathematical-process standards: require justification of claims, multiple representations of the same idea, attention to precision in units and labeling, and identification of patterns that support generalization.

Arithmetic sequences: a_n = a_1 + (n − 1)d; geometric sequences: a_n = a_1·r^(n−1). The common difference or ratio is found by subtracting or dividing consecutive terms.

Linear functions: slope m = (y_2 − y_1)/(x_2 − x_1); slope-intercept form y = mx + b; point-slope form y − y_1 = m(x − x_1); parallel lines share m, perpendicular lines have m_1·m_2 = −1.

Quadratic tools: vertex x = −b/(2a) for y = ax² + bx + c; discriminant b² − 4ac determines two, one, or zero real roots; factored form reveals x-intercepts and standard form reveals the y-intercept c.

Exponent and radical rules: a^m·a^n = a^(m+n), (a^m)^n = a^(mn), a^(−n) = 1/a^n, a^(1/n) = ⁿ√a; the domain of an even root requires a nonnegative radicand.

Function notation: f(g(x)) denotes composition, evaluated inside-out; the inverse f⁻¹ undoes f, so f(f⁻¹(x)) = x, and the graph of f⁻¹ reflects the graph of f across y = x.

Exponential growth/decay: y = a·b^x with b > 1 for growth and 0 < b < 1 for decay; compound interest A = P(1 + r/n)^(nt) and continuous growth A = Pe^(rt).

Systems of equations: substitution isolates one variable; elimination aligns coefficients so adding or subtracting removes a variable; a linear-quadratic system has 0, 1, or 2 real solutions depending on the discriminant.

Transformations: y = f(x − h) + k shifts the graph h units right and k units up; a·f(x) stretches vertically by |a| and reflects across the x-axis when a < 0.

Inequalities: multiplying or dividing by a negative reverses the inequality sign; the solution of a system is the overlap of the individual solution regions.

Perimeter and area: rectangle A = lw, triangle A = ½bh, parallelogram A = bh, trapezoid A = ½(b_1 + b_2)h, circle C = 2πr and A = πr².

Solid volume: prism/cylinder V = Bh, pyramid/cone V = ⅓Bh, sphere V = (4/3)πr³; surface area of a cylinder is 2πr² + 2πrh.

Pythagorean theorem a² + b² = c² holds for right triangles; the converse identifies right triangles, and common triples include 3-4-5 and 5-12-13.

Angle relationships: vertical angles are equal; complementary angles sum to 90° and supplementary to 180°; interior angles of an n-gon sum to (n − 2)·180°.

Similar figures have equal corresponding angles and proportional sides, with area ratio equal to the square of the scale factor and volume ratio equal to its cube.

Coordinate geometry: distance = √((x_2 − x_1)² + (y_2 − y_1)²); midpoint = ((x_1 + x_2)/2, (y_1 + y_2)/2); transformations preserve congruence when they are translations, rotations, or reflections.

Unit conversion uses dimensional analysis: multiply by conversion factors written as fractions so unwanted units cancel; 1 in = 2.54 cm and 1 km ≈ 0.621 mi.

Measurement precision: the precision of a computed result is limited by the least precise measurement, and relative error = |measured − actual| / actual.

Measures of center: mean = sum/n, median is the middle value of ordered data, mode is most frequent; the mean is pulled toward outliers while the median resists them.

Measures of spread: range = max − min; interquartile range IQR = Q3 − Q1; standard deviation measures typical distance from the mean and grows with spread.

Counting: fundamental counting principle multiplies independent choices; permutations P(n, r) = n!/(n − r)! order items, combinations C(n, r) = n!/(r!(n − r)!) do not.

Probability: P(E) = favorable outcomes / total equally likely outcomes; P(not E) = 1 − P(E); P(A or B) = P(A) + P(B) − P(A and B); independent events satisfy P(A and B) = P(A)·P(B).

Conditional probability: P(A|B) = P(A and B)/P(B); mutually exclusive events have P(A and B) = 0, and without replacement changes the denominator on later draws.

Expected value = Σ (value × probability); the binomial probability of exactly k successes in n trials is C(n, k)·p^k·(1 − p)^(n − k).

Graph theory: Euler's formula V − E + F = 2 for connected planar graphs; a graph has an Euler circuit when every vertex has even degree, and a Hamiltonian path visits each vertex once.

Sequences and series: arithmetic sum S_n = n(a_1 + a_n)/2; finite geometric sum S_n = a_1(1 − r^n)/(1 − r) for r ≠ 1; recursion defines a term from earlier terms plus a base case.

Integer operations: a − (−b) = a + b; (−a)(−b) = ab; (−a)(b) = −ab. Even and odd counts of negative factors determine the sign of a product.

Fraction operations: add or subtract only after finding a common denominator; multiply numerators and denominators directly; divide by multiplying by the reciprocal.

Scientific notation: a × 10^n with 1 ≤ |a| < 10; add exponents when multiplying powers of ten and subtract when dividing.

Rational vs. irrational: a rational number can be written as a ratio of integers; √2, π, and e are irrational, and the sum of a rational and an irrational is irrational.

Prime factorization supports greatest common factor (product of shared prime powers) and least common multiple (product of highest prime powers).

Ordering numbers: convert all to decimals or common denominators before comparing; on a number line, values increase to the right, so −7 < −2.

Estimation with benchmarks: 0.33 ≈ 1/3 and 0.67 ≈ 2/3; use these to check whether a computed fraction is plausible.

Pattern rules: identify whether consecutive differences are constant (linear), second differences constant (quadratic), or ratios constant (exponential).

Slope interpretation: positive slope rises left to right, negative falls, zero is horizontal, undefined is vertical.

Zero product property: if ab = 0 then a = 0 or b = 0; this solves factored quadratics such as (x − 3)(x + 5) = 0.

Solving linear equations: perform the same operation on both sides to isolate the variable, then substitute back to verify the solution.

Domain and range: domain lists allowable inputs, range lists resulting outputs; a function passes the vertical line test.

Logarithms undo exponentials: log_b(xy) = log_b x + log_b y and log_b(x^n) = n·log_b x; log_b 1 = 0.

Triangle congruence criteria: SSS, SAS, ASA, AAS, and HL for right triangles; AAA guarantees similarity, not congruence.

Circle facts: an inscribed angle is half its intercepted arc; a central angle equals its arc; a tangent is perpendicular to the radius at the point of tangency.

Arc length = (θ/360)·2πr and sector area = (θ/360)·πr², where θ is the central angle in degrees.

Special right triangles: 45-45-90 sides are in ratio 1:1:√2; 30-60-90 sides are in ratio 1:√3:2.

Slope criteria: parallel lines have equal slopes; perpendicular lines have slopes whose product is −1.

Composite figures: decompose into rectangles, triangles, and circles, sum the areas, and subtract any overlapping region counted twice.

Data displays: histograms show frequency of intervals, box plots show five-number summaries (min, Q1, median, Q3, max), and scatter plots reveal association.

Correlation coefficient r ranges from −1 to 1; |r| near 1 means strong linear association, and correlation does not imply causation.

Line of best fit predicts values; residuals are observed minus predicted, and a good fit has residuals scattered without pattern.

Sampling: random sampling reduces bias; larger samples reduce variability of estimates, and a sample statistic estimates a population parameter.

Venn diagrams and two-way tables support P(A or B) and conditional probability by counting cells; the total of all cells is the sample space size.

Recursive and explicit forms: a recursive rule needs a starting value plus a rule for later terms; an explicit rule computes any term directly from n.

Operations with decimals: align decimal points to add or subtract; the product of two decimals has as many decimal places as the sum of the factors' places.

Ratio and rate: a proportion a/b = c/d can be solved by cross multiplication; unit price = total cost ÷ quantity.

Rounding rules: look at the digit immediately right of the target place; 5 or greater rounds up, otherwise round down.

Number properties: commutative (a + b = b + a), associative (grouping changes), distributive a(b + c) = ab + ac, and identity elements 0 for addition and 1 for multiplication.

Percent problems: part = percent × whole; convert percent to decimal by dividing by 100 before multiplying.

Place-value comparison: the digit farthest left determines magnitude, so 0.7 > 0.65 because tenths outweigh hundredths.

Modeling with functions: identify the constant rate for linear models and the constant percent change for exponential models.

Arithmetic vs. geometric: arithmetic sequences add a fixed difference; geometric sequences multiply by a fixed ratio.

Graphing lines: plot the y-intercept b, then use slope m = rise/run to locate a second point.

Function composition and inversion: (f ∘ g)(x) = f(g(x)); find an inverse by swapping x and y and solving for y.

Absolute value functions y = |x − h| + k form a V with vertex at (h, k).

Quadratic vertex form y = a(x − h)² + k shows vertex (h, k) and opens upward when a > 0.

Area formulas: rhombus A = (d_1·d_2)/2; regular polygon A = ½·apothem·perimeter.

Surface area of a rectangular prism = 2(lw + lh + wh); volume = lwh.

Angle sum of a triangle is 180°; an exterior angle equals the sum of the two remote interior angles.

Scale drawings: map distance ÷ scale factor gives actual distance; keep units consistent.

Coordinate transformations: reflection across the x-axis maps (x, y) to (x, −y); across the y-axis maps to (−x, y); rotation 180° maps to (−x, −y).

Density and derived measures: density = mass/volume; speed = distance/time.

Mean absolute deviation: average of |each value − mean|; a larger value indicates greater spread.

Outlier rule: a value beyond Q1 − 1.5·IQR or Q3 + 1.5·IQR is often flagged as an outlier.

Permutation with repetition: arrangements of n items with repeated groups = n!/(n_1!·n_2!·…).

Probability of at least one success = 1 − P(no successes), useful for repeated independent trials.

Matrix and network basics: adjacency matrices record edges between vertices; the sum of a row gives that vertex's degree.

Fair games: a game is fair when expected value equals zero; positive expected value favors the player.

Signed fraction operations: keep the sign rules for integers and apply them to the numerator; −3/4 + 1/4 = −2/4 = −1/2.

Comparing fractions: cross multiply a/b vs. c/d; if ad > bc then a/b > c/d for positive denominators.

Exponent zero and one: a^0 = 1 for a ≠ 0, and a^1 = a.

Square roots: √(ab) = √a·√b and √(a/b) = √a/√b for nonnegative a and positive b.

Estimation check: multiplying numbers near 1 keeps the product near 1; numbers less than 1 shrink the product.

Precision in reporting: significant figures in a product match the factor with the fewest significant figures.

Slope from a table: divide the change in output by the change in input between any two rows; constant result confirms linearity.

Systems by graphing: the intersection point is the solution; parallel lines mean no solution, and coincident lines mean infinitely many.

Exponential vs. linear growth: linear adds a constant each step; exponential multiplies by a constant factor each step and eventually overtakes linear growth.

Domain restrictions: denominators cannot be zero, and even roots require nonnegative radicands.

Inequality graphing: a dashed boundary excludes the line (< or >); a solid boundary includes it (≤ or ≥).

Function families: linear, quadratic, exponential, absolute value, and rational each have characteristic graphs and rates of change.

Volume of a composite solid: add volumes of non-overlapping parts and subtract volumes of removed parts.

Circle inscribed in a square: the circle's diameter equals the square's side, so the circle's area is π(s/2)².

Midsegment of a triangle is parallel to the third side and half its length.

Similar triangles from parallel lines: corresponding angles are equal, so sides are proportional.

Angle of elevation and depression problems use right-triangle ratios: tan θ = opposite/adjacent.

Coordinate proof: verify parallel sides by equal slopes and right angles by negative reciprocal slopes.

Two-way tables: joint probability uses a single cell over the grand total; marginal probability uses a row or column total over the grand total.

Combinations with restriction: choose required members first, then fill remaining slots from the rest.

Expected value of a die: for a fair six-sided die, E = (1 + 2 + 3 + 4 + 5 + 6)/6 = 3.5.

Tree diagrams multiply probabilities along branches and add across branches for distinct outcomes.

Standard deviation vs. range: standard deviation uses every value; range uses only the extremes and is sensitive to outliers.

Recursive sequences in real contexts: compound interest is a geometric recursion with ratio (1 + r).