GACE® Mathematics (6–12) (711)
Practice Test

Estimation, computation, and number properties: using estimation strategically in a variety of situations — knowing when to estimate vs. compute exactly, and selecting appropriate estimation strategies (rounding to significant figures, front-end estimation, compatible numbers, clustering); solving problems involving integers (including negative integers and absolute value), rational numbers (fractions, mixed numbers), decimals, and percents — including conversions between these forms and applications such as finding percentages of quantities, percent increase/decrease, and discounts; solving problems involving ratios and proportions (unit rates, scaling, similar figures, maps, and other proportional contexts) and average rates of change (slope as the average rate of change of a function over an interval; average speed and other applied rates); translating among multiple mathematical representations — numerical, graphical, symbolic/algebraic, verbal, and tabular — and identifying which representation best illuminates a given mathematical relationship; identifying viable and non-viable arguments in mathematical reasoning — evaluating the logical validity of mathematical claims, proofs, and counterexamples; constructing and critiquing arguments about mathematical properties and procedures; solving complex multi-step problems that require sequencing multiple mathematical processes, managing intermediate computations, and attending to precision throughout.

Number systems, complex numbers, and number theory: analyzing the group structure of real numbers — understanding properties of number systems (closure, commutativity, associativity, identity, inverse, distributivity) and knowing which properties hold for which operations and number sets (for example: subtraction is closed on integers but not on natural numbers; division is not closed on integers); understanding the hierarchy of number sets: natural numbers ⊂ whole numbers ⊂ integers ⊂ rational numbers ⊂ real numbers, with irrational numbers (non-repeating, non-terminating decimals: √2, π, e) as the set completing the reals; using complex numbers and their operations — definition of imaginary unit i where i² = −1; complex numbers in the form a + bi (a = real part, b = imaginary part); adding/subtracting complex numbers (add/subtract real and imaginary parts separately); multiplying complex numbers using the distributive property and i² = −1; dividing complex numbers by multiplying numerator and denominator by the complex conjugate; the complex conjugate of a + bi is a − bi; modulus (absolute value) of a complex number: |a + bi| = √(a² + b²); the complex plane (Argand diagram); properties of numbers and operations including commutativity, associativity, distributivity, identity, and inverse elements; principles of basic number theory: prime and composite numbers; the Fundamental Theorem of Arithmetic (unique prime factorization); divisibility rules; GCF and LCM; the Euclidean algorithm; modular arithmetic (clock arithmetic); integer division and the division algorithm (a = qb + r); properties of even and odd numbers; perfect squares and perfect cubes; properties of zero (multiplication, division, identity).

Functions, polynomial functions, and their graphs: understanding functions as rules that assign exactly one output to each input; domain and range; function notation f(x); evaluating functions; determining whether a relation is a function (vertical line test; table analysis); linear functions (y = mx + b: slope-intercept form; slope as rate of change; y-intercept as starting value; graphing by plotting the y-intercept and using the slope; writing equations given two points or slope and one point; point-slope form; standard form; parallel lines [equal slopes] and perpendicular lines [slopes are negative reciprocals]; the relationship between a linear function and its graph); quadratic functions (y = ax² + bx + c or vertex form y = a(x−h)² + k; the parabola; vertex [h, k] is the maximum if a < 0, minimum if a > 0; axis of symmetry x = h; finding vertex by completing the square or using x = −b/(2a); x-intercepts [roots/zeros] by factoring, quadratic formula, or completing the square; discriminant b² − 4ac: positive → 2 real roots; zero → 1 real root; negative → 2 complex roots); higher-order polynomial functions (factoring: GCF, difference of squares a²−b²=(a+b)(a−b), sum/difference of cubes, factoring by grouping, synthetic division, the rational root theorem; the Fundamental Theorem of Algebra: a degree-n polynomial has exactly n complex roots counting multiplicity; end behavior; turning points).

Algebraic manipulation, systems, inverses, and advanced functions: manipulating algebraic expressions, equations, and inequalities — simplifying by combining like terms, distributing, and factoring; solving multi-step linear equations; solving and graphing linear inequalities (reverse inequality sign when multiplying/dividing by a negative); absolute value equations (|x| = a has solutions x = a and x = −a if a ≥ 0); solving systems of linear equations (substitution, elimination, Cramer’s rule, matrix methods) and linear inequalities (graphing half-planes; the solution region is the intersection); modeling application-based problems in Georgia contexts (interpreting slope and intercept in real situations; setting up and solving systems; linear programming); determining inverses and compositions of algebraic functions — the inverse function f⁻¹(x) undoes f(x): if f(a) = b then f⁻¹(b) = a; to find an inverse, replace f(x) with y, swap x and y, solve for y; composition f(g(x)) means apply g first, then f; properties of inverses and compositions; laws of exponents (aᵐ · aⁿ = aᵐ⁺ⁿ; aᵐ/aⁿ = aᵐ⁻ⁿ; (aᵐ)ⁿ = aᵐⁿ; a⁰ = 1 for a ≠ 0; a⁻ⁿ = 1/aⁿ) and logarithms (logᵇ(x) = y means bʸ = x; logᵇ(xy) = logᵇx + logᵇy; logᵇ(x/y) = logᵇx − logᵇy; logᵇ(xⁿ) = n·logᵇx; change-of-base formula: logᵇ(x) = ln(x)/ln(b)); exponential functions (y = a·bₓ: growth if b > 1, decay if 0 < b < 1; graph is always positive, passes through (0,a), has a horizontal asymptote at y = 0) and logarithmic functions (inverse of exponential; y = logᵇ(x): domain is x > 0, vertical asymptote at x = 0, passes through (1,0)); manipulating rational (polynomial/polynomial; identifying and simplifying; finding asymptotes), radical (√; fractional exponents: x¹⁄ⁿ = ⁿ√x; xᵐ⁄ⁿ = (ⁿ√x)ᵐ), and absolute value expressions and equations.

Measurement, units, similarity, and 3D figures: various units and unit conversions within and between the US customary system (12 in = 1 ft; 3 ft = 1 yd; 5,280 ft = 1 mi; 16 oz = 1 lb; 2,000 lb = 1 ton; 8 fl oz = 1 cup; 4 qt = 1 gal) and the metric system (km, m, cm, mm; kg, g, mg; L, mL; conversions using powers of 10); converting between customary and metric using conversion factors (1 in = 2.54 cm; 1 km ≈ 0.621 mi; 1 kg ≈ 2.2 lb; 1 L ≈ 1.057 qt); the concepts of similarity and scale factors (similar figures have equal corresponding angles and proportional corresponding sides; scale factor k; corresponding lengths scale by k, areas by k², volumes by k³; using proportions to find missing measures); proportional reasoning in scale drawings, maps, and models; precision, error, and rounding in measurements and computed quantities (significant figures; accuracy vs. precision; absolute error and relative error; effect of rounding on computed results); the concepts of perimeter, circumference (C = 2πr = πd), area (rectangle: lw; parallelogram: bh; triangle: ½bh; trapezoid: ½(b₁+b₂)h; circle: πr²; composite figures), surface area and volume of prisms (SA = 2B + Ph; V = Bh), cylinders (SA = 2πr² + 2πrh; V = πr²h), pyramids (SA = B + ½Pl; V = &frac13;Bh), cones (SA = πr² + πrl; V = &frac13;πr²h), and spheres (SA = 4πr²; V = &frac43;πr³); using nets (unfolded 2D representations) and cross sections (the shape produced by slicing a 3D figure with a plane) to analyze three-dimensional figures; effect of scaling on area (scale factor squared) and volume (scale factor cubed).

Euclidean and non-Euclidean geometry, proofs, coordinate geometry, and transformations: the axioms of Euclidean geometry (five postulates, particularly the parallel postulate: through a point not on a line, exactly one parallel line can be drawn); non-Euclidean geometries (spherical/elliptic geometry: the parallel postulate fails — no parallel lines exist on a sphere; hyperbolic geometry: infinitely many parallel lines through a point; both arise by modifying or negating the parallel postulate); properties of polygons (sum of interior angles of a polygon = (n−2)×180°; properties of parallelograms, rectangles, rhombuses, squares, trapezoids, kites; polygon angle sums) and circles (central angle, inscribed angle theorem [inscribed angle = ½ central angle]; arc length = (central angle/360)×2πr; sector area = (central angle/360)×πr²; tangent lines; chord-chord, secant-secant, and secant-tangent angle relationships; power of a point theorem); analyzing formal geometric proofs (two-column proofs, paragraph proofs, flow proofs; using properties, postulates, and theorems; proving triangles congruent: SSS, SAS, ASA, AAS, HL; proving triangles similar: AA, SAS, SSS similarity; CPCTC) and informal proofs; coordinate geometry (plotting points; distance formula d = √((x₂−x₁)²+(y₂−y₁)²); midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2); slope m = (y₂−y₁)/(x₂−x₁); using coordinate geometry to classify figures and prove geometric properties algebraically); performing transformations on figures in the coordinate plane: translations ((x,y)→(x+a,y+b)), reflections (over x-axis: (x,y)→(x,−y); over y-axis: (x,y)→(−x,y); over y=x: (x,y)→(y,x)), rotations (90° CCW: (x,y)→(−y,x); 180°: (x,y)→(−x,−y); 90° CW: (x,y)→(y,−x)), dilations ((x,y)→(kx,ky)); identifying lines of symmetry and rotational symmetry; compositions of transformations.

Trigonometry: functions, the unit circle, identities, and applications: the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) and their definitions using a right triangle (SOH-CAH-TOA: sin = opposite/hypotenuse; cos = adjacent/hypotenuse; tan = opposite/adjacent; csc = 1/sin; sec = 1/cos; cot = 1/tan) and using the unit circle (the unit circle is a circle of radius 1 centered at the origin; a point (x, y) on the unit circle at angle θ gives cosθ = x and sinθ = y); key unit circle values to know exactly: 0° (0, π), 30° (π/6), 45° (π/4), 60° (π/3), 90° (π/2) and their reference angle equivalents in all four quadrants; signs of trig functions by quadrant (All Students Take Calculus: Q1 all positive; Q2 sine positive; Q3 tangent positive; Q4 cosine positive); the domain, range, period, and graphs of sinθ (period 2π; range [−1, 1]), cosθ (period 2π; range [−1, 1]), and tanθ (period π; range all reals; vertical asymptotes at θ = π/2 + nπ); amplitude, period, phase shift, and vertical shift in y = A·sin(Bθ + C) + D; inverse trig functions (arcsin, arccos, arctan) and their restricted domains and ranges; trigonometric identities: Pythagorean identity (sin²θ + cos²θ = 1; tan²θ + 1 = sec²θ; 1 + cot²θ = csc²θ); reciprocal identities; quotient identities; even/odd identities; sum and difference formulas (sin(A ± B) = sinA cosB ± cosA sinB; cos(A ± B) = cosA cosB &mp; sinA sinB); double angle formulas; manipulating trig expressions and solving trig equations; laws of sines (a/sinA = b/sinB = c/sinC) and cosines (c² = a² + b² − 2ab·cosC) — used to solve oblique (non-right) triangles; the ambiguous case of the law of sines (SSA configuration can have 0, 1, or 2 solutions); applications: solving triangles, navigation, and surveying; analyzing conic sections algebraically: the circle (x−h)² + (y−k)² = r²; the ellipse (x−h)²/a² + (y−k)²/b² = 1 with a > b; the hyperbola (x−h)²/a² − (y−k)²/b² = 1; the parabola y−k = a(x−h)²; identifying each conic from its equation by the coefficients of x² and y².

Calculus: limits, derivatives, and integrals: evaluating limits with and without bound — limₓ→ₘ f(x) is the value f approaches as x approaches a (NOT necessarily the value at a); techniques for evaluating limits: direct substitution (if f is continuous at a), factoring and canceling (for 0/0 indeterminate forms), rationalizing, and the squeeze theorem; limits involving infinity: horizontal asymptotes (limₓ→∞ f(x) = L means the function approaches L as x grows without bound); points of discontinuity: removable discontinuities (holes — the limit exists but f(a) is not defined or does not equal the limit; can be “patched”), jump discontinuities (left-hand and right-hand limits exist but are not equal), and infinite discontinuities (vertical asymptotes; the function grows without bound); a function is continuous at a if: (1) f(a) is defined, (2) limₓ→ₘ f(x) exists, and (3) limₓ→ₘ f(x) = f(a); calculating derivatives of functions — the derivative f′(x) measures the instantaneous rate of change of f at x; definition of derivative as a limit: f′(x) = limₕ→₀ [f(x+h)−f(x)]/h; power rule: d/dx[xⁿ] = nxⁿ⁻¹; constant rule: d/dx[c] = 0; sum/difference rule; product rule: d/dx[fg] = f′g + fg′; quotient rule: d/dx[f/g] = (f′g−fg′)/g²; chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x); derivatives of trig functions (d/dx[sinx] = cosx; d/dx[cosx] = −sinx; d/dx[tanx] = sec²x); derivatives of exponential and logarithmic functions (d/dx[eₓ] = eₓ; d/dx[ln x] = 1/x); analyzing graphs of functions using derivatives: f′(x) > 0 means f is increasing; f′(x) < 0 means f is decreasing; f′(x) = 0 or undefined indicates a critical point (possible local maximum or minimum); f′′(x) > 0 means f is concave up; f′′(x) < 0 means f is concave down; inflection points where concavity changes; the First Derivative Test and Second Derivative Test for classifying critical points as local maxima, minima, or neither; solving optimization problems (find the critical points of an objective function, verify using the second derivative test, check endpoints); calculating integrals of functions — the indefinite integral ∫ f(x)dx = F(x) + C where F′(x) = f(x) (antiderivative); basic integration rules: power rule ∫xⁿdx = xⁿ⁺¹/(n+1) + C (n ≠ −1); the fundamental theorem of calculus: ∫ₘᵇ f(x)dx = F(b) − F(a); the definite integral as the net signed area under the curve; u-substitution (the chain rule in reverse).

Descriptive statistics, data visualization, and sampling: using visual representations to organize and display data: histograms (frequency distributions; equal-width bins; useful for showing the overall shape of a distribution; skewed left, skewed right, or symmetric/bell-shaped); box plots (five-number summary: minimum, Q1, median, Q3, maximum; IQR = Q3 − Q1; outlier fences: Q1 − 1.5×IQR and Q3 + 1.5×IQR); stem-and-leaf plots (show individual data values); dot plots; scatter plots (show relationship between two quantitative variables: positive association, negative association, no association; linear vs. nonlinear; the correlation coefficient r measures the strength and direction of a linear relationship: r = 1 perfect positive; r = −1 perfect negative; r = 0 no linear relationship); line of best fit (least-squares regression line &ycirc; = b₀ + b₁x; slope b₁ is the predicted change in y for a 1-unit increase in x; y-intercept b₀ is the predicted value of y when x = 0; residuals [observed − predicted] and residual plots); analyzing measures of central tendency (mean = arithmetic average [sum divided by n]; sensitive to outliers; median = the middle value when sorted; resistant to outliers; mode = most frequent value) and variability (range = max − min; IQR = Q3 − Q1; variance = average squared deviation from mean; standard deviation = square root of variance [measures typical distance from the mean]; coefficient of variation); analyzing the effects of bias in surveys and sampling: convenience sample (easily accessible members; likely biased), voluntary response sample (people self-select; usually biased toward extreme opinions), systematic sample (every kth member), stratified sample (divide into subgroups/strata; randomly sample from each — ensures representation), cluster sample (divide into clusters; randomly select entire clusters); why random sampling is essential for valid statistical inference; types of observational studies (surveys, retrospective, prospective) and controlled experiments (random assignment to treatment and control groups; key principle: randomization controls for confounding variables; the difference between causation and correlation).

Counting principles, probability, and distributions: counting principles: the fundamental counting principle (if event A can occur in m ways and event B in n ways, they can occur together in m×n ways); permutations (ordered arrangements of objects; P(n,r) = n!/(n−r)! — the number of ways to arrange r objects chosen from n without repetition); combinations (unordered selections; C(n,r) = n!/(r!(n−r)!) — the number of ways to choose r objects from n without regard to order; n choose r or binomial coefficient C(n,r)); the addition rule of probability: P(A or B) = P(A) + P(B) − P(A and B); for mutually exclusive events: P(A or B) = P(A) + P(B); the multiplication rule: P(A and B) = P(A)×P(B|A); for independent events: P(A and B) = P(A)×P(B); conditional probability P(A|B) = P(A and B)/P(B); Bayes’ theorem; determining probabilities of simple events (single outcome) and compound events (two or more events); complementary events P(A′) = 1 − P(A); geometric probability (favorable area / total area); expected value E(X) = Σx·P(X=x) (the long-run average value of a random variable); selecting simulations that model application-based events (using random number generators, dice, coins, or spinners to model probabilities empirically); analyzing probability distributions: uniform distribution (all outcomes equally likely; the probability histogram is flat); binomial distribution (n independent Bernoulli trials each with probability p of success; X ~ B(n,p); P(X=k) = C(n,k)×pᵏ×(1−p)ⁿ⁻ᵏ; mean μ = np; standard deviation σ = √(np(1−p))); normal distribution (symmetric, bell-shaped; characterized by mean μ and standard deviation σ; the empirical [68-95-99.7] rule: ~68% of values within 1σ of mean; ~95% within 2σ; ~99.7% within 3σ; z-score = (x − μ) / σ standardizes values for the standard normal distribution N(0,1); using z-tables to find probabilities).

Financial responsibility, decision-making, and planning: the principles of financial responsibility — understanding the long-term consequences of financial decisions; the importance of living within one’s means; the concept of a personal budget (income − expenses = surplus [save/invest] or deficit [spend from savings or borrow]); fixed vs. variable expenses; creating and maintaining a personal budget using the 50/30/20 rule (50% needs, 30% wants, 20% savings) or other frameworks; financial goal-setting using the SMART framework (Specific, Measurable, Achievable, Relevant, Time-bound); the importance of emergency savings (3–6 months of expenses as a liquid, easily accessible reserve); understanding the time value of money: a dollar today is worth more than a dollar in the future because of its earning potential; simple interest I = P×r×t (interest earned on principal only); compound interest A = P(1 + r/n)ⁿᵗ (interest earned on both principal and accumulated interest; more powerful than simple interest over time); the Rule of 72 (70/r ≈ years to double an investment at interest rate r); opportunity cost applies to financial decisions (using money for one purpose means forgoing its alternative uses); consumer rights and protections under federal law (Truth in Lending Act [TILA], Equal Credit Opportunity Act [ECOA], Fair Credit Reporting Act [FCRA], Fair Debt Collection Practices Act [FDCPA], Consumer Financial Protection Bureau [CFPB]).

Risk management, investing, and credit and debt: tools used to inform consumers and manage risk — insurance as a risk management tool: types of insurance (health insurance [premium, deductible, copayment, coinsurance, out-of-pocket maximum, HMO vs. PPO vs. HDHP], life insurance [term vs. whole life], auto insurance [liability, collision, comprehensive, uninsured motorist], homeowner’s/renter’s insurance, disability insurance); the basic principle of insurance: individuals pool risk; the insurer pays large unexpected losses in exchange for smaller predictable premiums; concepts of probability and expected value applied to insurance pricing; consumer tools: comparison shopping, reading and understanding contracts and warranty terms, researching before purchasing, understanding product safety standards; principles of investing — the relationship between risk and return (higher potential return requires accepting higher risk; the risk-return trade-off); types of investment vehicles: stocks (ownership shares in a corporation; potential for high returns and dividends but higher risk), bonds (debt instruments: the bond purchaser lends money to an issuer in exchange for periodic interest payments and return of principal at maturity; generally lower risk and lower return than stocks), mutual funds (pooled investment funds that hold diversified portfolios), index funds (passively track a market index; generally lower fees than actively managed funds), ETFs (exchange-traded funds: like index funds but traded on stock exchanges throughout the day), and real estate; diversification (spreading investments across asset classes, sectors, and geographies to reduce unsystematic risk); the importance of starting investing early (compound interest amplifies the earlier you begin); dollar-cost averaging (investing a fixed amount at regular intervals regardless of price); tax-advantaged retirement accounts (401(k), IRA, Roth IRA); managing credit and debt — what credit is (the ability to borrow money or access goods/services with the commitment to pay later); types of credit: revolving credit (credit cards: a credit limit, minimum monthly payment, and APR [Annual Percentage Rate]; carrying a balance incurs compound interest), installment credit (car loans, student loans, mortgages: fixed monthly payments over a defined term), open credit (utility accounts); credit scores and credit reports: the credit score (FICO score: 300–850) is calculated from five factors — payment history (35%: most important; always pay on time), amounts owed/credit utilization (30%: keep below 30% of available credit), length of credit history (15%), credit mix (10%), new credit (10%); the three major credit bureaus (Equifax, Experian, TransUnion) maintain credit reports; Americans are entitled to one free credit report per year from each bureau at AnnualCreditReport.com; strategies for managing and reducing debt: paying more than the minimum, targeting high-interest debt first (avalanche method), targeting smallest balance first (snowball method for psychological momentum), debt consolidation; understanding predatory lending practices (payday loans, rent-to-own, subprime loans) and how to avoid them.