Math

44 questions across two 35-minute modules. Calculator allowed on every question (built-in Desmos plus any personal calculator). About 75% multiple choice and 25% student-produced response (free-entry). Four content domains cover the same algebra-through-trig surface, with question difficulty escalating from Module 1 into the harder Module 2.

The four math domains

Algebra

The biggest domain (~35%). It's all linear: equations, inequalities, lines, and systems. Master the order-of-operations and translation basics first — most "hard" algebra questions are easy concepts wrapped in a long sentence.

Key moves

• Slope is everything. m = (y₂ − y₁)/(x₂ − x₁). Parallel ⇒ equal slopes; perpendicular ⇒ m₁ · m₂ = −1 (negative reciprocals).
• System solution count. Put both lines in y = mx + b. One solution ⇒ different slopes; no solution ⇒ same slope, different b; infinite solutions ⇒ same slope and same b (the equations are multiples). For a₁x + b₁y = c₁ and a₂x + b₂y = c₂: infinite ⇔ a₁/a₂ = b₁/b₂ = c₁/c₂; none ⇔ a₁/a₂ = b₁/b₂ ≠ c₁/c₂.
• Plug in the answers (PITA). When the algebra is messy but choices are numbers, test them — start at B or C since choices run least-to-greatest.
• Pick numbers when variables appear in the answer choices. Choose easy values, compute the target, match.
• Read the final question (RTFQ). The SAT often asks for an expression like 8x − 4, not for x. Solve for what's actually asked — solving for the variable and stopping is the #1 trap.
• Graph it in Desmos. A system's solution is the intersection point; an inequality's solution is the shaded overlap. Faster and safer than hand algebra.

Order of operations & expressions

• PEMDAS — Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).
• Distribute carefully: 5(2 − 4x) = 10 − 20x. Watch the sign when distributing a negative: −(3x − 2) = −3x + 2.
• FOIL to multiply binomials: (2x − 5)(3 + 4x) = 6x + 8x² − 15 − 20x = 8x² − 14x − 15.
• Combine like terms in polynomial add/subtract problems; write every step so a stray sign doesn't cost you.

Linear equations & inequalities

• Solve ax + b = c by isolating the variable: undo addition/subtraction first, then multiplication/division.
• Inequalities solve like equations, with one rule: flip the sign when you multiply or divide both sides by a negative. −2x + 4 > 6 ⇒ −2x > 2 ⇒ x < −1.
• Compound inequalities ("at least 1.5 and no more than 3.5") translate to 1.5 ≤ h ≤ 3.5.
• Translate words to math: "is" → =, "more than / increased by" → +, "less than / fewer than" → −, "of / per" → ×, "is at least" → ≥, "is at most" → ≤.

Absolute value

• |a| = distance from 0, always ≥ 0. |−6| = 6.
• Solve |x + 1| = 5 by splitting into two equations: x + 1 = 5 and x + 1 = −5, giving x = 4 or x = −6.
• If the absolute value equals 0 there is one solution; if it equals a negative number there are zero solutions.
• Absolute-value inequality |x − 4| ≤ 2 unfolds to −2 ≤ x − 4 ≤ 2 ⇒ 2 ≤ x ≤ 6.

Slope & line forms

• Slope between two points: m = (y₂ − y₁)/(x₂ − x₁) = rise over run.
• Slope-intercept y = mx + b: m = slope, b = y-intercept (value at x = 0).
• Point-slope y − y₁ = m(x − x₁) when you know a point and the slope.
• Standard form Ax + By = C: slope −A/B, x-intercept C/A, y-intercept C/B.
• Parallel ⇒ equal slopes; perpendicular ⇒ slopes are negative reciprocals (m₁·m₂ = −1); horizontal line slope 0, vertical line slope undefined.

Systems of linear equations

• Substitution: isolate one variable, plug into the other equation.
• Elimination: scale an equation so a variable cancels when you add. Often you can add/subtract to get the exact expression asked for (e.g. 7x − 8y) without solving for x and y separately.
• One solution ⇒ different slopes (lines cross once).
• No solution ⇒ same slope, different y-intercept (parallel, never meet).
• Infinite solutions ⇒ same slope and same intercept (one equation is a multiple of the other).
• In Desmos, the solution is the intersection point; use a k slider to find the constant that makes lines parallel (no solution) or identical (infinite).

Literal equations

• "Solve for P in terms of the other variables" — treat the target letter as the unknown and isolate it with the same moves you'd use on numbers.
• Example: from A = P(1 + r/n)^(nt), divide both sides by (1 + r/n)^(nt) to get P = A·(1 + r/n)^(−nt).

Advanced math

The other ~35%. Nonlinear: quadratics, higher polynomials, exponentials, radicals, rationals, and function behavior. The recurring skill is reading information directly off the right form of an equation instead of grinding algebra.

Key moves

• Read the parabola off its form. Standard ax² + bx + c ⇒ y-intercept is c; vertex a(x − h)² + k ⇒ vertex (h, k), axis x = h; factored a(x − r₁)(x − r₂) ⇒ roots r₁, r₂. The vertex x-coordinate is always x = −b/(2a).
• Quadratic formula & discriminant. x = (−b ± √(b² − 4ac))/(2a). The sign of b² − 4ac gives the number of real solutions: + ⇒ two, 0 ⇒ one (double root), − ⇒ none.
• Factor theorem. P(r) = 0 ⇔ (x − r) is a factor; a degree-n polynomial has at most n real roots, and the constant term is the product of the roots' opposites.
• Exponential form. y = a · bˣ — growth when b > 1, decay when 0 < b < 1; a is the starting value. "Doubles every k years" ⇒ y = a · 2^(t/k); "grows/loses r% per period" ⇒ b = 1 ± r/100.
• Function transformations. f(x) + k up, f(x) − k down; f(x + h) left, f(x − h) right; −f(x) flips over x-axis, f(−x) flips over y-axis.
• When stuck, graph or plug in. Drop the equation in Desmos and read intercepts, vertices, and intersections; or test the numeric answer choices directly. Desmos can also regress quadratic and exponential models from a table.

Factoring patterns

• Difference of squares: a² − b² = (a + b)(a − b).
• Perfect-square trinomials: a² + 2ab + b² = (a + b)²; a² − 2ab + b² = (a − b)².
• Common factor first: pull out the GCF before anything else (2x² + 6x = 2x(x + 3)).
• To factor x² + bx + c, find two numbers that multiply to c and add to b.

Quadratics

• Three forms: standard ax² + bx + c (reads off the y-intercept c), vertex a(x − h)² + k (reads off vertex (h, k)), factored a(x − r₁)(x − r₂) (reads off roots).
• Vertex / axis of symmetry: x = −b/(2a); parabola opens up if a > 0 (minimum), down if a < 0 (maximum).
• Solve by: factoring; square-root method when there's no bx term (x² = 49 ⇒ x = ±7); completing the square; or the quadratic formula.
• Completing the square also converts standard form to vertex form, and general circle form to center-radius form.
• Discriminant b² − 4ac: positive ⇒ two real roots, zero ⇒ one repeated root, negative ⇒ no real roots.
• Sum of roots = −b/a; product of roots = c/a.

Polynomials & the factor theorem

• A polynomial of degree n has at most n real roots and at most n − 1 turning points.
• Factor theorem: if P(r) = 0 then (x − r) is a factor, and the graph crosses (or touches) the x-axis at x = r.
• Match a polynomial's zeros to its graph's x-intercepts; a doubled factor (x − r)² means the graph touches but doesn't cross at r.
• Add, subtract, and multiply polynomials by combining like terms (watch signs across subtraction); divide via long or synthetic division.

Exponents, radicals & rational exponents

• Rules: aᵐ · aⁿ = aᵐ⁺ⁿ, aᵐ / aⁿ = aᵐ⁻ⁿ, (aᵐ)ⁿ = aᵐⁿ, (ab)ⁿ = aⁿbⁿ, a⁰ = 1, a⁻ⁿ = 1/aⁿ.
• Radicals as fractional exponents: √a = a^(1/2), ⁿ√a = a^(1/n), a^(m/n) = ⁿ√(aᵐ).
• Simplify radicals by pulling out perfect squares: √50 = 5√2.

Exponential growth & decay

• Model y = a · bˣ: a = starting amount, b = growth factor. Growth if b > 1, decay if 0 < b < 1.
• "Increases r% each period" ⇒ b = 1 + r/100; "decreases r%" ⇒ b = 1 − r/100.
• "Doubles every 3 years" ⇒ y = a · 2^(t/3); "halves every k periods" ⇒ y = a · (½)^(t/k).
• Compound interest: A = P(1 + r/n)^(nt).
• Linear vs. exponential: linear changes by a constant amount each step; exponential changes by a constant factor (percent) each step.

Rational & radical equations

• Rational expressions: factor numerator and denominator, cancel common factors, and note restrictions where the denominator = 0 (those make the function undefined).
• Clear fractions by multiplying through by the common denominator.
• Radical equations: isolate the radical, then square both sides — and always check for extraneous solutions that squaring can introduce.

Function notation, composition & graphs

• f(4) means substitute 4 for x: if f(x) = 3x − 5, then f(4) = 7. Remember f(x) is the y-value.
• Composition (f ∘ g)(x) = f(g(x)) — work inside-out. g(f(2)): find f(2) first, then feed that into g.
• Transformations: f(x) + k shifts up k; f(x) − k down; f(x + h) left h; f(x − h) right h; −f(x) reflects over the x-axis; f(−x) over the y-axis.
• Reading graphs: zeros / x-intercepts are where f(x) = 0; the y-intercept is f(0); count zeros by counting x-axis crossings.

Problem-solving & data analysis

Key moves

• Percent change & chaining. (new − old)/old × 100%. Successive changes multiply, not add: +20% then −10% ⇒ 1.20 × 0.90 = 1.08 (net +8%). "p% of x" ⇒ (p/100)·x.
• Center vs. spread. mean = sum/count, median = middle of sorted list, mode = most frequent. Adding a constant to every value shifts mean & median but leaves standard deviation unchanged; multiplying scales all three. Outliers move the mean far more than the median.
• Scatterplots & line of best fit. The line's slope = predicted change in y per unit x; its y-intercept = predicted y at x = 0. Use it to predict, and to spot which points lie above/below the trend. In Desmos, regression gives the equation instantly.
• Two-way tables. "Probability that…" ⇒ divide by the grand total; "given that / among the X" ⇒ divide by that row's or column's total only.
• Probability rules. P = favorable/total; independent ⇒ P(A and B) = P(A)·P(B); mutually exclusive ⇒ P(A or B) = P(A) + P(B).
• Unit conversion. Chain conversion factors so units cancel; carry units through every step as a built-in error check.

Ratios, proportions & rates

• A ratio compares parts; set a proportion a/b = c/d and cross-multiply to solve for the unknown.
• Rates: distance = rate × time; combine "per" quantities by multiplying or dividing as the units demand.
• Scale ratios up to totals: if a mix is 2 : 3 and there are 20 parts total, the pieces are 8 and 12.

Percentages & percent change

• "p% of x" = (p/100)·x. To find what percent a is of b: (a/b)·100%.
• Percent change: (new − old)/old × 100%. Increase ⇒ multiply by 1 + r/100; decrease ⇒ by 1 − r/100.
• Successive percent changes multiply, don't add: a 20% increase then a 10% decrease = 1.20 × 0.90 = 1.08 → net +8%.
• Ballpark trap: "30% shorter than 50" is 50 − 0.3·50 = 35, not 50 + 15 and not just 0.3·50.
• Simple interest I = Prt; compound interest A = P(1 + r/n)^(nt).

Unit conversion

• Chain conversion factors so unwanted units cancel diagonally; the unit you want ends up on top.
• Carry units through every step — if they don't cancel to the target unit, the setup is wrong. This is a built-in error check.
• Or set a proportion and cross-multiply: (known unit)/(known) = (target unit)/(x).

Statistics: center & spread

• Mean: sum ÷ count. Median: middle value when sorted (average the two middles if even count). Mode: most frequent. Range: max − min.
• Outliers drag the mean far more than the median; the median is the resistant measure of center.
• Standard deviation (conceptual) measures spread about the mean — tighter clustering ⇒ smaller SD; more spread ⇒ larger SD. You won't compute it by hand, but you'll compare two data sets.
• Adding a constant to every value shifts mean & median by that constant; range and SD are unchanged.
• Multiplying every value by a constant multiplies mean, median, range, and SD by that constant.

Graphs, scatterplots & tables

• Read carefully: note axis labels, units, and scale before answering. Bar graphs, histograms, line graphs, and box plots each show distribution differently.
• Scatterplots & line of best fit: the line's slope = predicted change in y per unit x; its y-intercept = predicted y at x = 0. Use the line to predict and to spot points above/below the trend. Desmos regression gives the equation instantly.
• Two-way tables: "probability that…" ⇒ divide by the grand total; "given that / among the X" ⇒ divide by that row's or column's total only.
• Interpreting constants in models: in y = mx + b from a word problem, m is the rate of change (per unit) and b is the starting/fixed value.

Probability

• P(event) = favorable/total (a number between 0 and 1).
• Independent events: P(A and B) = P(A)·P(B).
• Mutually exclusive events: P(A or B) = P(A) + P(B).
• Conditional ("given that X"): restrict to the X subgroup, then take favorable ÷ that subgroup total.

Sampling & margin of error

• Conclusions generalize only to the population the sample was randomly drawn from; a biased or non-random sample supports no valid conclusion.
• Margin of error gives a plausible range around the estimate; a larger random sample shrinks the margin of error and increases confidence.
• An experiment can support a cause-effect claim only with random assignment to groups; observational data shows association, not causation.

Geometry & trigonometry

Key moves

• Special right triangles save the most time: 45-45-90 sides 1 : 1 : √2; 30-60-90 sides 1 : √3 : 2. Memorize Pythagorean triples 3-4-5, 5-12-13, 8-15-17 and their multiples to skip the a² + b² = c² arithmetic.
• SOH-CAH-TOA. sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj; complementary angles give sin θ = cos(90° − θ). Radians: π = 180°.
• Similar triangles. Equal angles ⇒ corresponding sides in proportion — set up a ratio and cross-multiply. This is the workhorse for "find the missing length" figures.
• Area & volume. Triangle ½bh, circle πr², circumference 2πr; cylinder πr²h, cone ⅓πr²h, sphere 4⁄3πr³, pyramid ⅓·l·w·h. (On the on-screen reference sheet — but recall is faster.)
• Circles: arc & sector scale with the central angle. arc = 2πr·(θ/360°), sector area = πr²·(θ/360°). Equation (x − h)² + (y − k)² = r²; convert from general form by completing the square.
• Angle facts. Triangle angles sum to 180°; vertical angles equal; a transversal across parallels makes corresponding and alternate-interior angles equal.

Lines & angles

• Angles on a straight line are supplementary (sum 180°); around a point sum 360°; vertical angles (across an intersection) are equal.
• Complementary angles sum to 90°.
• Parallel lines cut by a transversal: corresponding angles equal, alternate-interior angles equal, same-side interior angles supplementary.

Triangles

• Interior angles sum to 180°; an exterior angle equals the sum of the two remote interior angles.
• Pythagorean theorem: a² + b² = c² (c = hypotenuse).
• Pythagorean triples to skip the arithmetic: 3-4-5, 5-12-13, 8-15-17, and any multiple (6-8-10, …).
• Special right triangles: 45-45-90 sides 1 : 1 : √2; 30-60-90 sides 1 : √3 : 2 (short leg : long leg : hypotenuse).
• Similar triangles: equal angles ⇒ corresponding sides in proportion — set a ratio and cross-multiply (the workhorse for missing-length figures). Congruent: same size and shape.
• Area = ½ · base · height; equilateral triangle of side s has area (√3/4)s².

Polygons & quadrilaterals

• Sum of interior angles of an n-sided polygon = (n − 2)·180°; each angle of a regular polygon = (n − 2)·180°/n.
• Rectangle area l · w; square area s², diagonal s√2; parallelogram area base · height; trapezoid area ½(b₁ + b₂)·h.
• Hierarchy: quadrilateral ⊃ parallelogram ⊃ rectangle ⊃ square.

Circles

• Area πr²; circumference 2πr = πd; diameter = 2r.
• Arc & sector scale with the central angle: arc = 2πr·(θ/360°), sector area = πr²·(θ/360°).
• Central vs. inscribed angle: a central angle equals its intercepted arc; an inscribed angle is half the central angle on the same arc.
• A tangent meets the circle at one point and is perpendicular to the radius drawn to that point.
• Equation: (x − h)² + (y − k)² = r² with center (h, k), radius r. Convert general form to this by completing the square.
• On the SAT π ≈ 3 is a fine ballpark; usually leave answers in terms of π.

Volume & surface area

On the on-screen reference sheet — memorize anyway:

• Box: l · w · h. Cube: s³.
• Cylinder: πr²h. Cone: (1/3)πr²h. Sphere: (4/3)πr³.
• Pyramid: (1/3) · (base area) · h.
• Surface area of a sphere: 4πr² (given if needed). Surface area of a box = sum of the six faces.

Right-triangle trigonometry

• SOH-CAH-TOA: sin θ = opp/hyp, cos θ = adj/hyp, tan θ = opp/adj.
• Cofunction (complementary-angle) identity: sin θ = cos(90° − θ) and cos θ = sin(90° − θ).
• Radians: π rad = 180°. Convert by proportion: degrees/180° = radians/π.
• For trig of an angle given in radians, just set the built-in calculator to radian mode.

Section strategy

• Pace. 35 minutes ÷ 22 questions ≈ 95 seconds each. Easier in Module 1; harder in Module 2 (especially the latter half).
• Use Desmos aggressively. Graph everything: systems → intersection point; quadratic roots → x-intercepts; inequality regions → shaded area. Faster and less error-prone than algebra by hand.
• Plug in answer choices when the algebra is messy and the choices are simple numbers. Usually start with B or C.
• Pick numbers for problems with variables in the answer choices. Pick easy ones (n = 2, n = 10), compute, match to a choice.
• Always answer. No penalty for wrong; a guess on a question you skipped has positive expected value.

Fill-ins (student-produced response)

Strategy: about 25% of math questions are fill-ins (SPRs) — you type the answer instead of choosing. They cover the same topics and sit in the same difficulty order; the only difference is the format, so don't fear them.