AP, SAT and ACT · Mathematics

SAT Math review: key skills

SAT Math review cards: linear equations and systems, quadratics, exponentials, functions, percentages, statistics, geometry and trig, plus calculator tactics.

38 cards5 topicsAbout 15 minutes a roundDigital SAT style math preparation (algebra, advanced math, data analysis, geometry and trigonometry)

Independent study material written by StudyBrix. AP and SAT are trademarks of College Board and ACT is a trademark of ACT, Inc.; neither organisation is involved with or has reviewed this deck.

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About this deck

This deck is for students preparing for the digital SAT Math section and for anyone who wants a fast review of high school algebra, data analysis and geometry. The math section has two adaptive modules; performance on the first module sets the difficulty of the second. Questions come from four content areas: algebra, advanced math, problem-solving and data analysis, and geometry and trigonometry. Most are multiple choice, and some ask you to enter your own answer. A calculator is allowed throughout the math section, and a graphing calculator is built into the testing app. The cards follow those four areas and finish with a few strategy cards.

You should know basic algebra: solving linear equations, working with fractions and exponents, and plotting points. Each card has a worked example with numbers checked; cover the example, solve it, and then compare. Time yourself on a few cards in a row, since speed with routine skills leaves more time for harder questions.

Study in short, frequent sessions and shuffle the tags once you know them, because the test mixes topics. Keep a list of the cards you miss and redo them two days later.

Common mistakes: answering a different question than the one asked (solving for x when it asks for 2x + 1); mishandling percentages, such as thinking a 20% rise followed by a 20% drop returns to the start; and forgetting to check radical or rational equations for extraneous solutions. Read units and labels on graphs carefully before computing.

Before you start: Grade 9 to 11 algebra and geometry: linear equations, exponents, fractions, basic graphing and the Pythagorean theorem.

All 38 cards

Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Algebra

Slope from two points: Rise over run: (y2 - y1)/(x2 - x1); it gives the constant rate of change of a line.Example: Through (2, 3) and (6, 11): slope (11 - 3)/(6 - 2) = 2.
Slope-intercept form: y = mx + b, where m is the slope and b is the y-intercept (the value of y when x = 0).Example: y = -3x + 7 crosses the y-axis at 7 and drops 3 for each 1 step right.
Linear model in context: A starting amount plus a constant rate times the input; the slope is the rate and the intercept is the starting value.Example: A plan costs $40 to join plus $15 per month: C = 40 + 15m, so 6 months cost $130.
Solving a system by elimination: Add or subtract equations (after scaling if needed) to remove one variable, solve for the other, then substitute back.Example: x + y = 10 and x - y = 4: adding gives 2x = 14, so x = 7 and y = 3.
System with no solution: Two linear equations whose lines are parallel: same slope, different intercepts.Example: 2x + 3y = 6 and 4x + 6y = k have no solution for any k other than 12.
System with infinitely many solutions: The two equations describe the same line; one is a multiple of the other.Example: 2x + 3y = 6 and 4x + 6y = 12 share every solution.
Linear inequality in context: An inequality that models a limit such as a budget; check whether boundary values are included.Example: Shirts at $12 and hats at $8 with at most $200: 12s + 8h <= 200.
Absolute value equation: |expression| = c (with c > 0) splits into expression = c or expression = -c.Example: |x - 3| = 5 gives x = 8 or x = -2.

Advanced math

Solving a quadratic by factoring: Write the equation as ax^2 + bx + c = 0, factor, and set each factor equal to zero.Example: x^2 - 4x - 5 = (x - 5)(x + 1) = 0, so x = 5 or x = -1.
Discriminant: b^2 - 4ac: positive means two real solutions, zero means one, negative means no real solutions.Example: x^2 + 6x + k has exactly one solution when 36 - 4k = 0, so k = 9.
Vertex form of a parabola: y = a(x - h)^2 + k has vertex (h, k); complete the square to reach this form.Example: x^2 + 6x + 5 = (x + 3)^2 - 4, vertex (-3, -4).
Sum and product of roots: For ax^2 + bx + c = 0, the roots add to -b/a and multiply to c/a.Example: 2x^2 - 8x + 6 = 0: sum 4, product 3 (roots 1 and 3).
Exponential growth model: y = a(1 + r)^t: a is the starting value and r the growth rate per period.Example: 500(1.04)^t after 3 years: 500 x 1.124864 = 562.43.
Exponential decay and half-life: y = a(1/2)^(t/h) when an amount halves every h time units; a decay rate r gives a(1 - r)^t.Example: 80 mg with a 6 hour half-life: after 18 hours, 80 x (1/2)^3 = 10 mg.
Rational exponents: x^(m/n) means the nth root of x raised to the m: x^(m/n) = (n-th root of x)^m.Example: 8^(2/3) = (cube root of 8)^2 = 4.
Extraneous solutions of radical equations: Squaring both sides can create false solutions, so substitute every answer back into the original equation.Example: sqrt(x + 3) = x - 3 gives x = 6 or x = 1; only 6 works.
Function composition: f(g(x)) means evaluate g first, then feed the result into f.Example: f(x) = 2x + 1, g(x) = x^2: f(g(3)) = f(9) = 19.
Graph transformations: f(x - h) + k shifts f right h and up k; -f(x) reflects over the x-axis; a f(x) stretches vertically by a.Example: y = (x - 2)^2 + 3 is y = x^2 moved right 2 and up 3.
Zeros of a factored polynomial: Each factor (x - r) gives a zero at x = r, which is an x-intercept of the graph.Example: p(x) = (x + 2)(x - 1)(x - 5) has zeros at -2, 1 and 5.

Problem-solving and data analysis

Proportion setup: Set two equal ratios with matching units in the same positions, then cross multiply.Example: 3 cups of flour for 4 servings: x/10 = 3/4, so 10 servings need 7.5 cups.
Unit conversion: Multiply by conversion factors written as fractions equal to 1 so unwanted units cancel.Example: 60 miles/hour x 5280 ft/mile / 3600 s/hour = 88 ft/s.
Percent change: (new - old)/old x 100; successive changes multiply, they do not add.Example: 80 to 92 is a 15% rise; +20% then -20% gives 1.2 x 0.8 = 0.96, a 4% net drop.
Mean versus median with outliers: The mean is pulled toward extreme values; the median is resistant.Example: Data 3, 4, 5, 6, 32: median 5, mean 10.
Comparing standard deviations: A data set whose values sit farther from the mean has a larger standard deviation; it measures spread, not centre.Example: Set A {9, 10, 11} has a smaller SD than set B {2, 10, 18}, though both means are 10.
Probability from a two-way table: Use the row or column named in the condition as the denominator.Example: Of 40 juniors, 12 play sports. P(plays sports given junior) = 12/40 = 0.3.
Inference from a sample: A random sample supports conclusions about its own population only; a margin of error gives a plausible range for the true value.Example: Sample mean 4.2 hours with margin 0.3: the population mean is plausibly between 3.9 and 4.5 hours.
Slope of a line of best fit: The predicted change in y for each one-unit increase in x, read in the context's units.Example: Predicted price = 1.5 x weight + 2 (in dollars and kg): each extra kg adds about $1.50.
Linear versus exponential growth: Linear growth adds the same amount each step; exponential growth multiplies by the same factor each step.Example: 5, 8, 11, 14 is linear (+3); 5, 10, 20, 40 is exponential (x 2).

Geometry and trigonometry

Volume of a cylinder: V = pi r^2 h, the base area times the height.Example: r = 3 and h = 10: V = 90 pi, about 282.7 cubic units.
30-60-90 triangle: Sides in the ratio 1 : sqrt(3) : 2, opposite the 30, 60 and 90 degree angles.Example: Short leg 5: hypotenuse 10, long leg 5 sqrt(3).
Similar figures and scale factor: If lengths scale by k, areas scale by k^2 and volumes by k^3.Example: Triangle sides tripled: area becomes 9 times as large.
Right triangle trigonometry: sin = opposite/hypotenuse, cos = adjacent/hypotenuse, tan = opposite/adjacent.Example: If sin A = 3/5 in a right triangle, the sides are 3, 4, 5, so cos A = 4/5.
Complementary angle identity: In a right triangle the two acute angles add to 90 degrees, so sin x = cos(90 - x).Example: sin 25 degrees = cos 65 degrees.
Arc length and radians: 180 degrees equals pi radians; arc length s = r theta when theta is in radians.Example: r = 6 and theta = pi/3: s = 2 pi.
Equation of a circle: (x - h)^2 + (y - k)^2 = r^2 has centre (h, k) and radius r; complete the square to find them.Example: x^2 + y^2 - 6x + 4y - 12 = 0 becomes (x - 3)^2 + (y + 2)^2 = 25: centre (3, -2), r = 5.
Interior angle sum of a polygon: (n - 2) x 180 degrees for a polygon with n sides.Example: A hexagon: 4 x 180 = 720 degrees.

Test strategy

Graphing to solve: Enter both sides of an equation or both equations of a system into the graphing calculator and read the intersection points.Example: Graph y = x^2 - 4 and y = 2x - 1: they meet where x^2 - 2x - 3 = 0, at x = -1 and x = 3.
Testing answer choices: Substitute each choice into the equation or condition, starting with a middle value, when solving directly is slow.Example: Which x makes 3x + 7 = 2x + 12? Try 5: 22 = 22.