Every Math concept in the middle school Science Bowl study set (grades 6-8), with a short definition for each. Read through them here, then drill them as flashcards or practise real Math questions.
Form ax² + bx + c = 0. Quadratic formula: x = (−b ± √(b²−4ac)) / 2a. Discriminant Δ = b²−4ac: Δ > 0 → two real roots; Δ = 0 → one real root; Δ < 0 → two complex roots. Vertex at x = −b/2a. Sum of roots = −b/a; product of roots = c/a.
log_b(x) = y ↔ bʸ = x. ln(x) = log_e(x). Rules: log(ab) = log a + log b; log(a/b) = log a − log b; log(aⁿ) = n·log a; log_b(1) = 0; log_b(b) = 1. Change of base: log_b(x) = ln(x)/ln(b). Inverse of exponential: b^(log_b(x)) = x.
SOH-CAH-TOA: sin = opp/hyp, cos = adj/hyp, tan = opp/adj. Pythagorean identity: sin²θ + cos²θ = 1. Other identities: 1 + tan²θ = sec²θ; 1 + cot²θ = csc²θ. Law of Sines: a/sinA = b/sinB. Law of Cosines: c² = a² + b² − 2ab·cosC. Radians: 180° = π rad.
Instantaneous rate of change. Power rule: d/dx[xⁿ] = nxⁿ⁻¹. Product: (uv)' = u'v + uv'. Quotient: (u/v)' = (u'v − uv')/v². Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x). d/dx[sin x] = cos x; d/dx[eˣ] = eˣ; d/dx[ln x] = 1/x. Critical points where f'(x) = 0 or undefined.
Antiderivative / area under curve. Power rule: ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1). Fundamental Theorem: ∫[a,b] f(x)dx = F(b) − F(a). Key: ∫eˣdx = eˣ + C; ∫(1/x)dx = ln|x| + C; ∫cos x dx = sin x + C; ∫sin x dx = −cos x + C. Integration by substitution and by parts.
P(A) = favorable / total. P(A and B) = P(A)·P(B) for independent events. P(A or B) = P(A) + P(B) − P(A∩B). Combinations C(n,r) = n!/[r!(n−r)!] — order doesn't matter. Permutations P(n,r) = n!/(n−r)! — order matters. Conditional: P(A|B) = P(A∩B)/P(B).
Arithmetic: aₙ = a₁ + (n−1)d; sum Sₙ = n(a₁+aₙ)/2. Geometric: aₙ = a₁·rⁿ⁻¹; sum Sₙ = a₁(1−rⁿ)/(1−r). Infinite geometric series: S = a₁/(1−r) if |r| < 1. Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂ (1,1,2,3,5,8,13,…). Telescoping series: most terms cancel.
i = √(−1); i² = −1; i³ = −i; i⁴ = 1. Form z = a + bi. Addition: add real and imaginary parts. Multiplication: FOIL (use i² = −1). Modulus |z| = √(a²+b²). Conjugate: z̄ = a − bi; z·z̄ = |z|². Euler: e^(iθ) = cosθ + i sinθ. Division: multiply by conjugate over conjugate.
Mean μ = Σx/n. Median: middle value. Mode: most frequent. Variance σ² = Σ(x−μ)²/n. Standard deviation σ = √(variance). Normal distribution: 68–95–99.7 rule (1σ, 2σ, 3σ). Correlation r ∈ [−1, +1]. Linear regression: line of best fit. Outliers can skew mean but not median.
Rectangular arrays of numbers. Matrix multiplication: rows × columns (NOT commutative). det of 2×2 [a b; c d] = ad − bc. Inverse A⁻¹ exists iff det ≠ 0. Identity matrix I: A·I = A. Solve systems via Gaussian elimination or Cramer's Rule. Eigenvalue λ: Av = λv (eigenvector v).
Quantities with magnitude and direction. Component form: v = ⟨a, b, c⟩. Magnitude |v| = √(a²+b²+c²). Dot product: u·v = |u||v|cosθ = Σuᵢvᵢ (scalar; zero if perpendicular). Cross product: u×v is perpendicular to both; |u×v| = |u||v|sinθ. Unit vector: v/|v|.
lim_{x→a} f(x) = L if f(x) approaches L as x → a (from both sides). L'Hôpital's Rule: for 0/0 or ∞/∞ indeterminate forms, lim f/g = lim f'/g'. Continuity at a: f(a) defined, limit exists, and lim_{x→a} f(x) = f(a). Squeeze theorem: if g(x) ≤ f(x) ≤ h(x) and g,h → L, then f → L.
Prime numbers have exactly two factors: 1 and themselves. Fundamental Theorem of Arithmetic: every integer > 1 has a unique prime factorization. GCD: greatest common divisor; LCM: least common multiple; GCD × LCM = product of two numbers. Modular arithmetic: a ≡ b (mod n) means n divides (a−b). Euler's totient function φ(n).
Circle: (x−h)² + (y−k)² = r². Ellipse: x²/a² + y²/b² = 1 (a > b; foci at ±c where c² = a²−b²). Parabola: y = ax² + bx + c; focus/directrix definition. Hyperbola: x²/a² − y²/b² = 1; asymptotes y = ±(b/a)x. All are cross-sections of a double cone.
Long division of polynomials mirrors long division of integers. Synthetic division: shortcut when dividing by (x − c). Remainder Theorem: dividing f(x) by (x − c) gives remainder f(c). Factor Theorem: (x − c) is a factor of f(x) iff f(c) = 0. Rational Root Theorem: rational roots of aₙxⁿ + … + a₀ are ±(factors of a₀)/(factors of aₙ).
Domain: set of valid inputs. Range: set of outputs. Composition: (f∘g)(x) = f(g(x)). Inverse: f⁻¹ exists if f is one-to-one (passes horizontal line test); f(f⁻¹(x)) = x. Transformations: f(x) + k (vertical shift), f(x + h) (horizontal shift), af(x) (vertical stretch), f(bx) (horizontal compression).
(a + b)ⁿ = Σ C(n,k) aⁿ⁻ᵏ bᵏ for k = 0 to n. The kth term (starting from 1) = C(n,k−1)·aⁿ⁻⁽ᵏ⁻¹⁾·bᵏ⁻¹. Pascal's triangle rows give the coefficients. C(n,k) = n!/[k!(n−k)!]. Used in probability distributions (binomial distribution) and approximations.
Set: unordered collection of distinct elements. Union A ∪ B: elements in A or B. Intersection A ∩ B: elements in both. Complement Aᶜ: elements not in A. Empty set ∅. Subset: A ⊆ B if every element of A is in B. |A ∪ B| = |A| + |B| − |A ∩ B| (inclusion-exclusion). Venn diagrams visualize set relationships.
Optimize a linear objective function subject to linear inequality constraints. Feasible region: all points satisfying all constraints (convex polygon). Optimal solution always occurs at a vertex (corner point) of the feasible region. Used in operations research, resource allocation, and supply chain optimization.
Equations involving a function and its derivatives. Separable ODEs: dy/dx = f(x)g(y) → separate and integrate. Exponential growth/decay: dy/dt = ky → y = y₀eᵏᵗ. Second-order linear ODEs model springs (y'' + ky = 0 → y = A cos√k·t + B sin√k·t). Initial conditions determine particular solutions.
Points described by (r, θ): r = distance from origin, θ = angle from positive x-axis. Conversion: x = r cosθ, y = r sinθ; r = √(x²+y²), θ = arctan(y/x). Circle r = a. Rose curves: r = a cos(nθ). Cardioid: r = a(1 + cosθ). Area in polar: A = ½∫r²dθ.
f(x) = p(x)/q(x) where p, q are polynomials. Vertical asymptote where q(x) = 0 (and p(x) ≠ 0). Horizontal asymptote: if deg(p) < deg(q) → y = 0; deg(p) = deg(q) → y = ratio of leading coefficients; deg(p) > deg(q) → oblique asymptote. Holes where both p and q have a common factor.
|x| = x if x ≥ 0; −x if x < 0. |x| < a means −a < x < a. |x| > a means x > a or x < −a. Triangle inequality: |a + b| ≤ |a| + |b|. Solving quadratic inequalities: factor, find roots, test intervals. Sign chart method: track sign changes at roots and asymptotes.
Curve defined by x = f(t) and y = g(t) where t is a parameter. Eliminate the parameter to get Cartesian form. Derivative: dy/dx = (dy/dt)/(dx/dt). Arc length: L = ∫√[(dx/dt)²+(dy/dt)²] dt. Circle: x = r cosθ, y = r sinθ. Projectile: x = v₀cosθ·t, y = v₀sinθ·t − ½gt².
Graphs consist of vertices (nodes) and edges. Degree of vertex: number of edges connected. Euler path: visits every edge exactly once (exists if exactly 0 or 2 odd-degree vertices). Euler circuit: Euler path starting and ending at same vertex (0 odd-degree vertices). Trees: connected graphs with no cycles; n vertices, n−1 edges.
Taylor series: f(x) = Σ f⁽ⁿ⁾(a)/n! · (x−a)ⁿ; approximates a function around point a. Maclaurin series: Taylor series centered at 0. Common Maclaurin series: eˣ = 1 + x + x²/2! + x³/3! + …; sin x = x − x³/3! + x⁵/5! − …; cos x = 1 − x²/2! + x⁴/4! − …; ln(1+x) = x − x²/2 + x³/3 − … (|x| ≤ 1). Radius of convergence determined by ratio test. Truncating gives polynomial approximations.
U-substitution: reverse of chain rule; let u = g(x), du = g'(x)dx. Integration by parts: ∫u dv = uv − ∫v du (choose u by LIATE: Logarithm, Inverse trig, Algebraic, Trig, Exponential). Trigonometric substitution: for √(a²−x²) use x = a sinθ; for √(a²+x²) use x = a tanθ; for √(x²−a²) use x = a secθ. Partial fractions: decompose rational functions into simpler fractions before integrating. Tabular integration: efficient for repeated integration by parts.
Functions of multiple variables f(x, y, z). Partial derivatives: ∂f/∂x holds other variables constant. Gradient ∇f = (∂f/∂x, ∂f/∂y, ∂f/∂z): points in direction of steepest ascent. Directional derivative: rate of change in direction of unit vector û. Double integrals: ∫∫f(x,y) dA; can switch to polar coordinates. Triple integrals: volume and mass calculations. Line integrals: ∫F·dr along a curve. Green's, Stokes', and divergence theorems connect line/surface/volume integrals.
Direct proof: assume hypothesis, show conclusion follows logically. Proof by contrapositive: prove 'not Q → not P' instead of 'P → Q'. Proof by contradiction: assume negation of conclusion, derive contradiction. Mathematical induction: base case (n=1 true) + inductive step (if n=k true, then n=k+1 true) → true for all positive integers. Strong induction: assume all cases up to k. Proof by cases: exhaust all possible cases. Counterexample: single example that disproves a universal claim.
Discrete distributions: Binomial P(X=k) = C(n,k)pᵏ(1−p)ⁿ⁻ᵏ; μ = np, σ² = np(1−p). Poisson P(X=k) = e^(−λ)λᵏ/k!; μ = σ² = λ (rare events in interval). Continuous distributions: Normal (bell curve; 68-95-99.7 rule). Exponential: time between Poisson events; memoryless property. Uniform: equal probability on interval. Central Limit Theorem: sample means approach normal distribution regardless of population distribution as n → ∞.
Base-2 (binary): digits 0,1; used in computers. Base-8 (octal): digits 0–7. Base-16 (hexadecimal): digits 0–9 and A–F. Conversion: repeatedly divide by base, collect remainders. Modular arithmetic: a ≡ b (mod n) means n | (a−b). Arithmetic mod n: (a+b) mod n, (ab) mod n. Fermat's Little Theorem: if p prime, aᵖ ≡ a (mod p). Chinese Remainder Theorem: system of congruences with coprime moduli has unique solution mod product. Applications: cryptography, hashing, calendar calculations.
Inclusion-exclusion principle: |A∪B∪C| = |A|+|B|+|C| − |A∩B| − |A∩C| − |B∩C| + |A∩B∩C|. Stars and bars: distributing n identical objects into k distinct bins = C(n+k−1, k−1). Derangements: permutations with no fixed point; D(n) = n!(1 − 1/1! + 1/2! − … + (−1)ⁿ/n!). Catalan numbers: Cₙ = C(2n,n)/(n+1); count valid parenthesizations and paths. Pigeonhole principle: if n+1 items in n containers, at least one container has ≥2 items.
Euler's formula: e^(iθ) = cosθ + i sinθ. Euler's identity: e^(iπ) + 1 = 0. De Moivre's theorem: (cosθ + i sinθ)ⁿ = cos(nθ) + i sin(nθ). Polar form of complex number: z = r e^(iθ) = r(cosθ + i sinθ). Multiplication in polar form: multiply moduli, add arguments. Roots of unity: solutions to zⁿ = 1 are equally spaced on unit circle. nth roots: n solutions, spaced 2π/n apart. Fourier series uses complex exponentials as basis functions.
Eigenvalue equation: Av = λv (A is matrix, λ is eigenvalue, v is eigenvector). Find eigenvalues: solve det(A − λI) = 0 (characteristic polynomial). Eigenvectors: solve (A − λI)v = 0. Diagonalization: A = PDP⁻¹ where D is diagonal matrix of eigenvalues, P has eigenvectors as columns. Symmetric matrices have real eigenvalues and orthogonal eigenvectors. Applications: principal component analysis (PCA), differential equations, Google PageRank, quantum mechanics.
Equations defining sequences in terms of previous terms. Linear recurrences with constant coefficients: aₙ = c₁aₙ₋₁ + c₂aₙ₋₂ + … Characteristic equation method: substitute aₙ = rⁿ to get characteristic polynomial; roots r give solution. Fibonacci: aₙ = aₙ₋₁ + aₙ₋₂; closed form (Binet's formula): Fₙ = (φⁿ − ψⁿ)/√5 where φ = (1+√5)/2 (golden ratio). Generating functions: represent sequence as power series coefficients to solve recurrences.
Defined using exponentials: sinh x = (eˣ − e⁻ˣ)/2; cosh x = (eˣ + e⁻ˣ)/2; tanh x = sinh x / cosh x. Identity: cosh²x − sinh²x = 1. Derivatives: d/dx[sinh x] = cosh x; d/dx[cosh x] = sinh x. Inverse: arcsinh x = ln(x + √(x²+1)); arccosh x = ln(x + √(x²−1)) (x ≥ 1). Applications: catenary curve (hanging chain); special relativity (rapidity); solutions to Laplace's equation.
Mathematical study of strategic decision-making. Nash equilibrium: no player can improve their payoff by unilaterally changing strategy; can have multiple or no pure-strategy equilibria. Prisoner's dilemma: individually rational choices lead to collectively suboptimal outcomes. Dominant strategy: best regardless of opponent's choice. Zero-sum games: one player's gain = other's loss (solved by minimax theorem). Mixed strategies: probabilistic choices. Applications: economics, evolutionary biology (evolutionarily stable strategies), political science.
Improper integral: infinite limits or discontinuous integrand. Type 1 (infinite limits): ∫[a,∞) f(x)dx = lim_{t→∞} ∫[a,t] f(x)dx. Type 2 (discontinuity): ∫[a,b] f(x)dx where f is discontinuous at endpoint. Comparison test: if 0 ≤ f(x) ≤ g(x) and ∫g converges, so does ∫f. p-integral: ∫[1,∞) x^(−p)dx converges iff p > 1. Gamma function: Γ(n) = ∫[0,∞) x^(n−1)e^(−x)dx = (n−1)! for positive integers.
Boolean algebra uses values 0 (false) and 1 (true). Operations: AND (A·B), OR (A+B), NOT (A'). De Morgan's Laws: (A·B)' = A'+B'; (A+B)' = A'·B'. Logic gates: AND, OR, NOT, NAND, NOR, XOR (A⊕B = A'B + AB'). Truth tables enumerate all input combinations. Sum of products (SOP) and product of sums (POS) forms. Karnaugh maps simplify Boolean expressions. Applications: digital circuit design, computer architecture, cryptography.
Algorithms for approximating solutions to mathematical problems. Newton-Raphson method: xₙ₊₁ = xₙ − f(xₙ)/f'(xₙ); fast convergence (quadratic) for root-finding. Bisection method: bracket root, halve interval; slower but guaranteed. Euler's method: numerical ODE solving; yₙ₊₁ = yₙ + h·f(xₙ, yₙ). Runge-Kutta (RK4): more accurate ODE solver. Trapezoidal rule and Simpson's rule: numerical integration. Error analysis: truncation error (method) vs roundoff error (finite precision).
Fraction: part of a whole (numerator/denominator). To convert fraction to decimal: divide numerator by denominator. Percent: parts per hundred; convert to decimal by dividing by 100 (75% = 0.75). Percent of a number: multiply (30% of 80 = 0.30 × 80 = 24). Operations with fractions: to add/subtract, find a common denominator; to multiply, multiply straight across; to divide, multiply by the reciprocal.
A ratio compares two quantities (3:4 or 3/4). A proportion is an equation stating two ratios are equal (3/4 = 6/8). Cross-multiplication: if a/b = c/d then ad = bc. Unit rate: ratio with denominator 1 (60 miles per hour). Scale factor: ratio used in scale drawings and similar figures. Direct variation: y = kx; as x increases, y increases proportionally.
Variable: a letter representing an unknown value. Expression: combination of numbers, variables, and operations (3x + 2). Equation: two expressions set equal (3x + 2 = 11). Solving: use inverse operations to isolate the variable. Order of operations: PEMDAS (Parentheses, Exponents, Multiplication/Division, Addition/Subtraction). Distributive property: a(b + c) = ab + ac.
Perimeter: sum of all side lengths. Area formulas: rectangle = length × width; triangle = ½ × base × height; parallelogram = base × height; trapezoid = ½(b₁ + b₂) × h; circle = πr². Circumference of circle = 2πr = πd. Volume: rectangular prism = l × w × h; cylinder = πr²h; cone = ⅓πr²h; sphere = 4/3πr³. Surface area = sum of all face areas.
Exponent rules: xᵃ × xᵇ = xᵃ⁺ᵇ; xᵃ / xᵇ = xᵃ⁻ᵇ; (xᵃ)ᵇ = xᵃᵇ; x⁰ = 1; x⁻ⁿ = 1/xⁿ. Scientific notation: a × 10ⁿ where 1 ≤ a < 10. To multiply in scientific notation: multiply the coefficients and add exponents. To add/subtract: first make the exponents equal. Examples: 3×10⁴ × 2×10³ = 6×10⁷.
Probability = number of favorable outcomes / total number of outcomes. Range: 0 (impossible) to 1 (certain). Complementary events: P(A) + P(not A) = 1. Independent events: P(A and B) = P(A) × P(B). Mutually exclusive events: P(A or B) = P(A) + P(B). Experimental probability: based on trials; theoretical probability: based on perfect conditions.
Mean (average): sum of values ÷ number of values. Median: middle value when data is ordered (average of two middle values if even count). Mode: most frequently occurring value. Range: maximum − minimum. Interquartile range (IQR): Q3 − Q1. Outlier: a data point that is much higher or lower than most. Bar graphs, histograms, box plots, and scatter plots display data.
Integers include positive whole numbers, negative whole numbers, and zero. Adding integers: same sign → add and keep sign; different signs → subtract and keep the sign of the larger absolute value. Multiplying/dividing: same signs → positive result; different signs → negative result. Absolute value |x|: distance from zero on a number line; always non-negative.
Linear equation: y = mx + b (slope-intercept form). Slope (m) = rise/run = (y₂ − y₁)/(x₂ − x₁); represents rate of change. Y-intercept (b): where line crosses the y-axis. Positive slope: line goes up left to right. Negative slope: line goes down. Horizontal line: slope = 0. Vertical line: undefined slope. Two lines are parallel if they have equal slopes.
The value of a digit depends on its position (place) in a number. In base 10: ones, tens, hundreds, thousands, … to the left of the decimal; tenths, hundredths, thousandths, … to the right. Expanded form: 3,472 = 3×1000 + 4×100 + 7×10 + 2×1. Rounding: look at the digit to the right of the rounding place; if ≥5, round up; if <5, round down. Comparing decimals: align decimal points and compare digit by digit. The number line shows all real numbers in order; negative numbers are to the left of zero.
Negative numbers are less than zero; used for temperatures below freezing, debt, sea level depths. On a number line, negatives are left of zero. Absolute value |x|: the distance from zero; always non-negative. |-5| = |5| = 5. Adding: same signs → add and keep sign; different signs → subtract and keep sign of larger absolute value. Subtracting: add the opposite. Multiplying/Dividing: same signs → positive; opposite signs → negative. Ordering integers: −10 < −3 < 0 < 5 < 12.
PEMDAS (Please Excuse My Dear Aunt Sally): Parentheses first, then Exponents, then Multiplication and Division (left to right), then Addition and Subtraction (left to right). Also called BODMAS (Brackets, Orders, Division/Multiplication, Addition/Subtraction). Example: 3 + 4 × 2² − (8 ÷ 4) = 3 + 4 × 4 − 2 = 3 + 16 − 2 = 17. Common mistakes: forgetting that multiplication and division have equal priority (left to right), and same for addition and subtraction. Brackets/parentheses always evaluated first, innermost first.
Factor: a number that divides evenly into another (factors of 12: 1, 2, 3, 4, 6, 12). Multiple: result of multiplying a number by an integer (multiples of 5: 5, 10, 15, 20…). GCF (Greatest Common Factor): largest factor shared by two numbers (GCF of 12 and 18 is 6). LCM (Least Common Multiple): smallest number divisible by both (LCM of 4 and 6 is 12). GCF × LCM = product of the two numbers. Divisibility rules: divisible by 2 (even), 3 (digit sum divisible by 3), 4 (last two digits), 5 (ends in 0 or 5), 6 (divisible by 2 and 3), 9 (digit sum divisible by 9), 10 (ends in 0).
Two perpendicular number lines (x-axis horizontal, y-axis vertical) intersecting at the origin (0,0). Points written as ordered pairs (x, y). Four quadrants: I (+, +), II (−, +), III (−, −), IV (+, −). Plotting points: move x units horizontally, y units vertically from origin. Distance between two points: d = √[(x₂−x₁)² + (y₂−y₁)²]. Midpoint: ((x₁+x₂)/2, (y₁+y₂)/2). Reflections: across x-axis → (x, −y); across y-axis → (−x, y); across origin → (−x, −y). Used for graphing equations and geometric transformations.
Inequality symbols: < (less than), > (greater than), ≤ (less than or equal), ≥ (greater than or equal), ≠ (not equal). Solving: same as equations, BUT when multiplying or dividing both sides by a negative number, flip the inequality sign. Graphing on a number line: open circle for < or > (endpoint not included); closed circle for ≤ or ≥ (included). Compound inequalities: AND (intersection; 'between' — both conditions must be met); OR (union; either condition). Example: −2 < x ≤ 5 means x is greater than −2 AND less than or equal to 5.
In a right triangle, the square of the hypotenuse (longest side, opposite the right angle) equals the sum of the squares of the other two legs: a² + b² = c². Used to find the length of any side if two sides are known. Example: a=3, b=4 → c=√(9+16)=√25=5 (3-4-5 is a Pythagorean triple). Other common triples: 5-12-13, 8-15-17. Also used to determine if a triangle is right-angled: if a² + b² = c², it's a right triangle. Converse: if a² + b² ≠ c², not a right triangle. Can find distances in the coordinate plane.
Complementary angles: sum to 90°. Supplementary angles: sum to 180°. Vertical angles: formed by intersecting lines; they are equal. Adjacent angles: share a vertex and a side. Straight angle = 180°. Right angle = 90°. Acute angle < 90°. Obtuse angle: between 90° and 180°. Transversal crossing parallel lines creates: corresponding angles (equal), alternate interior angles (equal), alternate exterior angles (equal), co-interior/consecutive interior angles (supplementary = 180°). Triangle angle sum = 180°. Exterior angle of triangle = sum of two non-adjacent interior angles.
Classified by sides: equilateral (all sides equal, all angles 60°), isosceles (two equal sides, two equal base angles), scalene (no equal sides). Classified by angles: acute (all angles < 90°), right (one 90° angle), obtuse (one angle > 90°). Triangle inequality: sum of any two sides > third side. Area = ½ × base × height. Perimeter = sum of all three sides. Similar triangles: same shape, proportional sides (AA, SAS, SSS similarity). Congruent triangles: same shape and size (SSS, SAS, ASA, AAS congruence). Largest angle opposite longest side.
A proportional relationship: y = kx where k is the constant of proportionality (unit rate). On a graph: a straight line through the origin. k = y/x for any point. Unit rate: ratio expressed as a quantity per one unit (speed in miles per hour, price per pound, heart beats per minute). Scale factor: ratio of corresponding measurements in similar figures. To solve proportions: cross-multiply. Direct proportion: as x increases, y increases proportionally. Inverse proportion: as x increases, y decreases (xy = k). Percent proportion: part/whole = percent/100.
Percent change = (new − old) / old × 100%. Percent increase: positive change. Percent decrease: negative change (% decrease = (old − new)/old × 100%). Simple interest: I = Prt (principal × rate × time in years). Discount: subtract percent of original price. Markup: add percent of cost to cost. Tax/tip: add percent of price. Finding original amount after percent change: original = new amount / (1 ± percent as decimal). Sales tax: total = price × (1 + tax rate). Example: item costs $80, 25% off: discount = $20, sale price = $60.
Volume: rectangular prism = l × w × h; cube = s³; cylinder = πr²h; cone = ⅓πr²h; sphere = (4/3)πr³; pyramid = ⅓ × base area × height. Surface area: rectangular prism = 2(lw + lh + wh); cube = 6s²; cylinder = 2πr² + 2πrh; sphere = 4πr². Net: a flat shape that can be folded into a 3D figure; used to visualize and calculate surface area. Cavalieri's Principle: two solids with equal height and equal cross-sectional areas at every level have equal volumes. Units of volume: cm³, m³, liters (1 L = 1000 cm³).
Translation: slides a figure along a vector (a, b) — adds a to x-coordinates, b to y-coordinates. Rotation: turns a figure around a center point by a given angle (90°, 180°, 270° CW or CCW). Reflection: flips a figure over a line of reflection. Dilation: enlarges or reduces a figure by a scale factor k (k>1 enlargement; 0<k<1 reduction). Rigid transformations (isometries): translations, rotations, reflections — preserve size and shape. Non-rigid: dilations — preserve shape but not size. Congruent figures: related by rigid transformations. Similar figures: related by dilations (and rigid transformations).
Two or more equations solved simultaneously. Graphing method: find intersection point of lines. Substitution method: solve one equation for a variable, substitute into the other. Elimination (addition) method: add or subtract equations to eliminate a variable. Three possible outcomes: one solution (lines intersect at a point — consistent and independent), no solution (parallel lines — inconsistent), infinitely many solutions (same line — consistent and dependent). Applications: finding two unknown quantities simultaneously (mixture problems, age problems, distance-rate-time problems).
A function: a rule that assigns exactly one output (y) to each input (x). Represented as f(x). Vertical line test: if any vertical line crosses the graph more than once, it's NOT a function. Domain: set of all valid inputs (x-values). Range: set of all outputs (y-values). Ways to represent functions: tables, graphs, equations, mapping diagrams. Linear function: f(x) = mx + b. Quadratic function: f(x) = ax² + bx + c (parabola). Increasing: output increases as input increases. Decreasing: output decreases as input increases.
Bar graph: compares categories using bars of different heights. Line graph: shows change over time; good for trends. Pie chart (circle graph): shows parts of a whole as sectors (percentages must sum to 100%). Histogram: bar chart for continuous data organized in intervals/bins; no gaps between bars. Scatter plot: shows relationship between two quantitative variables; each point represents one data pair. Line of best fit (trend line): drawn through scatter plot to show general trend. Positive correlation: both variables increase together. Negative correlation: one increases, other decreases. No correlation: no pattern.
Measures of center: mean (sum ÷ count; affected by outliers), median (middle value when ordered; resistant to outliers), mode (most frequent value; can have multiple or none). Measures of spread: range (max − min), interquartile range IQR = Q3 − Q1 (middle 50% of data; resistant to outliers), mean absolute deviation (MAD = average distance from mean). Standard deviation: average distance from the mean (larger → more spread out). Outlier: a value more than 1.5 × IQR below Q1 or above Q3. Box plot shows min, Q1, median, Q3, max.
Scientific notation: a × 10ⁿ where 1 ≤ a < 10. Very large: 4,300,000 = 4.3 × 10⁶. Very small: 0.00057 = 5.7 × 10⁻⁴. To multiply: multiply coefficients and add exponents (3×10⁴ × 2×10³ = 6×10⁷). To divide: divide coefficients and subtract exponents. To add/subtract: must convert to same power of 10 first. Applications in science: speed of light = 3×10⁸ m/s; mass of electron = 9.11×10⁻³¹ kg; distance to Andromeda galaxy = 2.5×10⁶ light-years. Calculators display scientific notation as '6.02E23' for 6.02×10²³.
Simple interest: I = Prt; total amount A = P(1 + rt). P = principal, r = annual interest rate (decimal), t = time (years). Example: $1000 at 5% for 3 years: I = 1000 × 0.05 × 3 = $150; total = $1150. Compound interest: interest is calculated on principal AND previously earned interest. Formula: A = P(1 + r/n)^(nt) where n = number of times compounded per year. Continuously compounded: A = Pe^(rt). Compound interest grows faster than simple interest. Rule of 72: approximate years to double money = 72 / interest rate (%).
Square root: √x = y means y² = x. Perfect squares: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225. √72 = 6√2 (simplify by factoring out largest perfect square). Cube root: ∛x = y means y³ = x. Perfect cubes: 1, 8, 27, 64, 125, 216, 343. Irrational numbers: square roots of non-perfect squares (√2 ≈ 1.414, √3 ≈ 1.732); cannot be expressed as a fraction; non-terminating, non-repeating decimals. Estimating square roots: √50 is between √49=7 and √64=8, closer to 7 since 50 is closer to 49.
Similar figures: same shape, different size; corresponding angles are equal; corresponding sides are proportional. Scale factor: ratio of corresponding sides (e.g., scale factor of 3 means all lengths tripled, areas × 9, volumes × 27). To find missing side in similar figures: set up proportion (a/b = c/d) and cross-multiply. Scale drawings: maps, blueprints, models use a scale ratio (e.g., 1 cm = 50 km). Indirect measurement: use similar triangles to measure heights/distances you cannot measure directly (e.g., shadow method to find tree height).
Compound event: involves two or more simple events. Independent events: outcome of one does not affect the other (flipping two coins). P(A and B) = P(A) × P(B) for independent events. Dependent events: outcome of one affects the other (drawing cards without replacement). P(A and B) = P(A) × P(B|A) (conditional probability). P(A or B) = P(A) + P(B) − P(A and B) (addition rule). Complementary: P(not A) = 1 − P(A). Sample space: list all possible outcomes. Tree diagrams and tables help organize compound probability problems.
Arithmetic sequence: each term found by adding a constant (common difference d). aₙ = a₁ + (n−1)d. Example: 3, 7, 11, 15 (d=4). Sum of n terms: Sₙ = n(a₁+aₙ)/2. Geometric sequence: each term found by multiplying by a constant (common ratio r). aₙ = a₁ × r^(n−1). Example: 2, 6, 18, 54 (r=3). Sum of geometric series: Sₙ = a₁(1−rⁿ)/(1−r). Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13 (each term is sum of previous two). Recognizing patterns: look for common differences, ratios, or more complex rules.
Circle: set of all points equidistant from center. Radius (r): center to edge. Diameter (d = 2r): across center. Circumference C = 2πr = πd. Area A = πr². π ≈ 3.14159. Arc: part of the circumference; arc length = (central angle/360°) × 2πr. Sector: pie-slice region; area = (central angle/360°) × πr². Chord: segment connecting two points on circle. Tangent: line touching circle at exactly one point. Inscribed angle: vertex on circle; inscribed angle = ½ × intercepted arc. Central angle = intercepted arc. Semicircle: 180°.
Rational numbers: can be expressed as a fraction p/q where p and q are integers and q ≠ 0; includes all integers, fractions, terminating decimals (0.75), and repeating decimals (0.333… = 1/3). Irrational numbers: cannot be expressed as a fraction; non-terminating, non-repeating decimals (√2 = 1.41421…; π = 3.14159…). Real numbers: all rational and irrational numbers combined. Converting repeating decimal to fraction: let x = 0.333…; 10x = 3.333…; 10x − x = 3; 9x = 3; x = 1/3. √(perfect square) is rational; √(non-perfect square) is irrational.
Goal: isolate the variable using inverse operations. Steps: 1) Use distributive property to remove parentheses. 2) Combine like terms on each side. 3) Use addition/subtraction to get variable terms on one side. 4) Use multiplication/division to solve for variable. Variables on both sides: collect all variable terms on one side, constants on the other. Example: 3(x + 2) = 2x + 11 → 3x + 6 = 2x + 11 → x = 5. Always check your answer by substituting back into the original equation. Equations with no solution (0=5) or infinite solutions (5=5).
Scale drawing: represents a real object proportionally. Scale: written as ratio (1:100) or with units (1 cm = 10 m). To find actual size: multiply drawing measurement by scale factor. To find drawing size: divide actual size by scale factor. Unit conversions (customary): 12 inches = 1 foot; 3 feet = 1 yard; 5,280 feet = 1 mile. Unit conversions (metric): kilo (×1000), hecto (×100), deka (×10), base, deci (÷10), centi (÷100), milli (÷1000). Converting metric: move decimal point. Converting between systems: 1 inch = 2.54 cm; 1 kg ≈ 2.2 lb; 1 mile ≈ 1.61 km.
Population: entire group being studied. Sample: subset of population used to make inferences. Random sample: every member has an equal chance of being selected — reduces bias. Sampling methods: simple random (lottery/random number), stratified (divide into subgroups, sample proportionally), systematic (every nth person), convenience (easiest available — most biased). Bias: systematic error that skews results. Sample size: larger samples → more accurate estimates. Margin of error: indicates how much sample results may differ from true population value. Surveys, experiments, and observational studies are different ways to collect data.