0% of the question bank attempted

Linear Equations in One Variable

Isolate the variable using inverse operations, applying the same operation to both sides to keep the equation balanced.
  • Combine like terms and distribute first: a(x+b) = ax+ab
  • Move variable terms to one side, constants to the other
  • Divide by the coefficient last to solve for x
  • Check your answer by substituting back into the ORIGINAL equation
  • Multiplying/dividing an inequality by a negative flips the inequality sign
  • Watch for equations with variables on both sides: 3x+5=2x-1 → x=-6

Systems of Linear Equations

A system's solution is the point (or points) where all equations are simultaneously true — solve by substitution, elimination, or graphing.
  • Substitution: solve one equation for a variable, plug into the other
  • Elimination: multiply equations so a variable's coefficients cancel when added/subtracted
  • Graphically, the solution is the intersection point of the two lines
  • Parallel lines (same slope, different intercept) → no solution
  • Identical lines → infinitely many solutions
  • 'How many solutions' questions test slope/intercept comparison, not actual solving

Linear Word Problems & Modeling

Translate real-world scenarios into equations: identify the rate (slope) and starting value (y-intercept).
  • y = mx + b: m is rate of change per unit, b is the initial/starting amount
  • 'Per', 'each', 'every' signal a rate (slope); a flat one-time amount is the intercept
  • Total cost = (unit price)(quantity) + fixed fee
  • For 'break-even' problems, set two total-cost expressions equal and solve
  • Units matter: convert to matching units before writing the equation

Linear Inequalities

Inequalities behave like equations except the sign flips when multiplying or dividing by a negative number.
  • Solve like an equation, flipping the inequality sign only when multiplying/dividing by a negative
  • Compound inequalities: solve all three parts together, e.g., -3 < 2x+1 < 7
  • 'At least' means ≥; 'at most' means ≤; 'more than' means >; 'fewer than' means <
  • Graph solutions on a number line: open circle for < or >, closed circle for ≤ or ≥
  • Word problems with constraints often need an inequality, not an equation

Interpreting Linear Graphs

The slope and intercept of a line carry real-world meaning that the SAT tests directly.
  • Slope = rise/run = (y2-y1)/(x2-x1); represents rate of change
  • y-intercept (x=0) is the starting value in context
  • x-intercept (y=0) is where the quantity reaches zero
  • Steeper slope = faster rate of change (positive) or faster decrease (negative)
  • Two lines with the same slope are parallel; slopes that multiply to -1 are perpendicular

Absolute Value Equations

|x| represents distance from 0, so absolute value equations split into two cases.
  • |x - a| = b means x is exactly b units from a: x = a+b or x = a-b
  • Isolate the absolute value expression before splitting into two equations
  • Always check both solutions in the original equation — extraneous roots can appear
  • |expression| = negative number has NO solution
Forgetting to flip the inequality sign when multiplying/dividing by a negative — always flip it, not just when the answer 'looks wrong.'
Distributing incorrectly: -2(x - 3) = -2x + 6, not -2x - 6.
Answering with the value of the wrong variable — the question may ask for y after you solved for x.
Assuming parallel lines intersect somewhere; same slope + different intercept = no solution, always.

Ratios, Rates, and Proportions

A ratio compares two quantities; a proportion sets two ratios equal and is solved by cross-multiplication.
  • Set up ratios with matching units in the same position: a/b = c/d
  • Cross-multiply: a·d = b·c
  • Unit rate = quantity ÷ number of units (e.g., $/item, miles/hour)
  • Scale factor problems: multiply every dimension by the same ratio
  • Watch for 'part-to-part' vs. 'part-to-whole' ratios — they're not interchangeable

Percentages

Percent means 'per hundred'; convert to a decimal (divide by 100) before computing.
  • Percent change = (new - old)/old × 100%
  • Percent increase: multiply by (1 + rate); percent decrease: multiply by (1 - rate)
  • 'X is what percent of Y?' → (X/Y) × 100
  • Successive percent changes do NOT add directly — apply them one after another
  • 'Percent of' means multiply; 'percent more/less than' means adjust the base by that percent

Mean, Median, and Standard Deviation

Measures of center describe a 'typical' value; spread measures describe how varied the data is.
  • Mean = sum of values ÷ number of values
  • Median = middle value when data is sorted (average of two middle values if even count)
  • Mode = most frequent value
  • Larger standard deviation = data is more spread out from the mean
  • An outlier pulls the mean toward it much more than the median

Reading Tables, Scatterplots & Two-Way Tables

Extract exact values from tables and describe trends/associations from scatterplots.
  • Two-way tables: row/column totals must match the grand total — use this to check your reading
  • Conditional relative frequency = cell value ÷ row or column total (read the question carefully for which)
  • Positive association: as x increases, y tends to increase (points trend upward)
  • Line of best fit summarizes a linear trend; use it to predict values not directly on the graph
  • Outliers on scatterplots don't fit the overall trend

Unit Conversion & Rates

Use dimensional analysis: multiply by conversion fractions so unwanted units cancel.
  • Set up conversion factors as fractions equal to 1: 1 mile/5280 feet
  • Cancel units diagonally like multiplying fractions
  • Rate problems: distance = rate × time, so time = distance/rate
  • Combined/average rate for a round trip is NOT the simple average of the two speeds — use total distance ÷ total time

Probability

Probability = favorable outcomes ÷ total possible outcomes, always between 0 and 1.
  • P(event) = favorable outcomes / total outcomes
  • For 'and' with independent events, multiply probabilities
  • For 'or' with mutually exclusive events, add probabilities
  • From a two-way table, P(A|B) = (count in both A and B) / (total count in B)
  • Probabilities from data tables use the given sample, not a formula
Using the wrong 'original' value in a percent change problem — always divide by the STARTING value, not the ending one.
Averaging two speeds directly for a round trip instead of using total distance ÷ total time.
Confusing mean and median — outlier-heavy data shifts the mean, not necessarily the median.
Forgetting that successive percent changes compound rather than add (a 10% increase then 10% decrease is NOT back to the original).

Factoring Quadratics

Rewrite ax²+bx+c as a product of binomials to reveal roots quickly.
  • For x²+bx+c, find two numbers that multiply to c and add to b
  • Difference of squares: a² - b² = (a+b)(a-b)
  • Perfect square trinomial: a² ± 2ab + b² = (a ± b)²
  • When a ≠ 1, use the 'ac method' or the quadratic formula
  • Once factored, set each factor equal to 0 to find the roots

The Quadratic Formula & Discriminant

x = (-b ± √(b²-4ac)) / 2a solves any quadratic ax²+bx+c=0.
  • Discriminant Δ = b² - 4ac tells you the number/type of roots
  • Δ > 0: two distinct real roots
  • Δ = 0: one repeated real root (vertex touches x-axis)
  • Δ < 0: no real roots (graph doesn't cross the x-axis)
  • Vertex x-coordinate: x = -b/2a

Exponent Rules

Exponent rules let you simplify expressions without expanding them.
  • Product rule: x^a · x^b = x^(a+b)
  • Quotient rule: x^a / x^b = x^(a-b)
  • Power rule: (x^a)^b = x^(ab)
  • Negative exponent: x^(-a) = 1/x^a
  • Zero exponent: x^0 = 1 (x ≠ 0)
  • Fractional exponent: x^(1/n) = ⁿ√x

Exponential Growth & Decay

Exponential functions model repeated percent change: y = a(1 ± r)^t.
  • a = initial amount, r = rate as a decimal, t = time periods
  • Growth: y = a(1+r)^t; decay: y = a(1-r)^t
  • Exponential growth eventually outpaces any linear or quadratic growth
  • The base (1+r) or (1-r) must be positive
  • Half-life/doubling problems use y = a(1/2)^(t/half-life) or y = a(2)^(t/doubling-time)

Function Notation & Transformations

f(x) notation describes input-output relationships; shifting the graph changes the equation predictably.
  • f(a) means substitute a for x in the function rule
  • f(x) + k shifts the graph up k units; f(x) - k shifts down
  • f(x + k) shifts left k units; f(x - k) shifts right k units
  • -f(x) reflects over the x-axis; f(-x) reflects over the y-axis
  • Composite functions: f(g(x)) means plug g(x) in for x in f

Systems with a Quadratic and a Line

A line and a parabola can intersect at 0, 1, or 2 points — set the equations equal to find them.
  • Set y = mx+b equal to y = ax²+bx+c, then solve the resulting quadratic
  • Two real solutions = two intersection points
  • One real solution = the line is tangent to the parabola
  • No real solutions = the line never touches the parabola
  • Substitute x-solutions back into the linear equation to get y-coordinates
Forgetting the ± in the quadratic formula — a quadratic almost always has two roots, not one.
Mixing up growth and decay: (1+r) grows, (1-r) decays — using the wrong sign inverts the whole model.
Sign errors when factoring: (x-3)(x+3) = x²-9, not x²+9.
Applying exponent rules to terms that are ADDED, not multiplied — x^a + x^b does NOT simplify to x^(a+b).

Area and Perimeter

Know the standard area/perimeter formulas for common shapes cold — the SAT rarely gives them for basic figures.
  • Rectangle: A = lw; Perimeter = 2(l+w)
  • Triangle: A = (1/2)bh
  • Circle: A = πr²; Circumference = 2πr
  • Trapezoid: A = (1/2)(b1+b2)h
  • Parallelogram: A = bh

The Pythagorean Theorem

In any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
  • a² + b² = c²: legs squared sum to the hypotenuse squared
  • Common Pythagorean triples: 3-4-5, 6-8-10, 5-12-13, 8-15-17
  • Distance formula IS the Pythagorean theorem: d = √[(x2-x1)²+(y2-y1)²]
  • Works ONLY for right triangles

Circles: Arcs, Sectors, and Angles

A circle has 360° total; arcs and sectors are fractional parts of the whole circle.
  • Arc length = (central angle/360°) × 2πr
  • Sector area = (central angle/360°) × πr²
  • Inscribed angle = half the central angle that subtends the same arc
  • A tangent line is perpendicular to the radius at the point of tangency
  • Equation of a circle: (x-h)² + (y-k)² = r², center (h,k), radius r

Volume and Surface Area (3D Solids)

3D formulas extend 2D area formulas by adding a height/depth dimension.
  • Rectangular prism: V = lwh
  • Cylinder: V = πr²h
  • Cone: V = (1/3)πr²h
  • Sphere: V = (4/3)πr³
  • Surface area of a cylinder: SA = 2πr² + 2πrh

Similar Triangles & Congruence

Similar figures have proportional corresponding sides and equal corresponding angles.
  • AA (Angle-Angle) similarity: two equal angles guarantee similar triangles
  • Corresponding sides of similar triangles are in the same ratio (scale factor)
  • If scale factor is k, areas scale by k² and volumes scale by k³
  • Congruent triangles: SSS, SAS, ASA, AAS prove exact congruence (not just similarity)

Angle Relationships

Parallel lines cut by a transversal create predictable, testable angle pairs.
  • Vertical angles are equal
  • A straight line totals 180°; angles around a point total 360°
  • Corresponding angles (parallel lines) are equal
  • Alternate interior angles (parallel lines) are equal
  • Co-interior (same-side interior) angles are supplementary (sum to 180°)
  • Triangle interior angles always sum to 180°
Using the diameter instead of the radius in area/circumference formulas — always confirm which one is given.
Forgetting that similar triangles scale AREA by k², not by k — a doubled side length means 4× the area.
Mislabeling the hypotenuse — it's always the side opposite the right angle, and the longest side.
Confusing arc length (a length, uses 2πr) with sector area (a region, uses πr²).

SOH-CAH-TOA

In a right triangle, trig ratios relate an acute angle to the ratio of two sides.
  • sin(θ) = opposite/hypotenuse
  • cos(θ) = adjacent/hypotenuse
  • tan(θ) = opposite/adjacent = sin(θ)/cos(θ)
  • 'Opposite' and 'adjacent' are defined relative to the angle θ you're using
  • Only valid for the acute angles of a RIGHT triangle

Special Right Triangles

The 45-45-90 and 30-60-90 triangles have fixed, memorizable side ratios.
  • 45-45-90: legs are equal, hypotenuse = leg × √2
  • 30-60-90: sides are in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse)
  • Short leg (opposite 30°) : long leg (opposite 60°) : hypotenuse (opposite 90°) = 1 : √3 : 2
  • These let you find exact side lengths without a calculator

Complementary Angles: sin/cos Relationship

In a right triangle, the two acute angles are complementary (sum to 90°), which links sine and cosine directly.
  • sin(θ) = cos(90° - θ)
  • If sin(A) = cos(B) and A, B are acute angles of the same right triangle, then A + B = 90°
  • This lets the SAT ask you to find an angle given a sine/cosine relationship without a triangle drawn
  • Also true: cos(θ) = sin(90° - θ)

The Unit Circle (Basics)

The unit circle extends trig ratios to all angles using a circle of radius 1 centered at the origin.
  • A point on the unit circle at angle θ has coordinates (cos θ, sin θ)
  • sin²θ + cos²θ = 1 for any angle θ (Pythagorean identity)
  • Radians: 180° = π radians, so to convert degrees to radians multiply by π/180
  • sin and cos values repeat every 360° (2π radians)

Solving for Missing Sides/Angles with Trig

Use trig ratios and inverse trig functions to find unknown sides or angles in right triangles.
  • Identify which two sides (relative to the known angle) are involved: opposite, adjacent, hypotenuse
  • Set up the correct ratio (sin, cos, or tan) and solve algebraically
  • To find an angle from a ratio, use inverse trig: sin⁻¹, cos⁻¹, tan⁻¹
  • Always double-check your calculator is in the correct mode (degrees vs. radians) if using inverse trig numerically
Mixing up which side is 'opposite' vs. 'adjacent' — it depends entirely on which angle you're referencing.
Using degree-mode formulas with a calculator set to radians (or vice versa) — always check the mode.
Forgetting that the hypotenuse is always the LONGEST side and is never labeled 'opposite' or 'adjacent.'
Assuming SOH-CAH-TOA works on non-right triangles — it only applies to right triangles.

When (and When Not) to Use Your Calculator

The calculator section allows a calculator, but the fastest solution is often mental math or estimation, not typing every step.
  • Use the calculator for messy arithmetic, decimals, and multi-step computations
  • Skip the calculator for simple factoring or obvious simplifications — typing takes longer than solving
  • Estimate first to sanity-check the calculator's output against a ballpark answer
  • Graphing calculators can check systems of equations by finding intersection points visually
  • Don't re-enter an entire expression from scratch after one small error — edit it

Solving Systems Graphically

A graphing calculator finds intersection points instantly instead of solving algebraically.
  • Graph both equations and use the 'intersect' feature to find solution points
  • Verify the intersection makes sense with the problem context (e.g., positive values only for real-world quantities)
  • For no solution: lines never cross (parallel)
  • For infinite solutions: the graphs completely overlap

Using Statistical Functions

Calculators can compute mean, median, and standard deviation directly from a list of data.
  • Enter data into a list, then use 1-Var Stats (or equivalent) to get mean (x̄) and standard deviation (σx)
  • Sort the list first if you need the median manually
  • Regression tools (LinReg) find the line/curve of best fit for scatterplot data
  • Double check you're reading σx (sample) vs σx (population) consistently with what's asked

Table Feature for Function Evaluation

The table (TABLE) feature evaluates a function at many x-values at once — faster than plugging in one at a time by hand.
  • Enter the function, then generate a table of x/y pairs
  • Useful for checking multiple answer choices against a function quickly
  • Helps visually spot where a function crosses zero (roots) or matches a target value
  • Change the table step size to zoom in near a suspected answer

Answer-Choice Elimination (Backsolving)

On calculator-allowed problems, plugging answer choices back into the problem is often faster than solving algebraically.
  • Start with a middle answer choice (like C) — if too big/small, you know which direction to go
  • Backsolving works great for equations with a single variable and messy coefficients
  • For 'which of the following' geometry problems, plug values into the given formula directly
  • Combine backsolving with estimation to eliminate obviously wrong choices first
Blindly trusting calculator output without an estimate to sanity-check — a single mistyped digit can produce a wildly wrong (but confident-looking) answer.
Re-typing a whole expression instead of editing after a typo, wasting valuable time.
Using calculator mode inconsistently (degrees vs. radians) for trig problems.
Forgetting order of operations when typing complex fractions — use extra parentheses around the entire numerator and denominator.

Mental Math Shortcuts

The no-calculator section rewards quick, exact mental arithmetic and smart simplification over brute-force computation.
  • Break numbers apart: 98 × 6 = (100-2) × 6 = 600 - 12 = 588
  • Memorize squares up to 20² and common fraction/decimal/percent equivalents (1/4=25%=0.25)
  • Simplify fractions before multiplying, not after
  • Look for common factors to cancel before doing heavy multiplication

Factoring to Simplify Fast

Factoring often turns an intimidating no-calculator expression into a one-step answer.
  • Pull out the GCF (greatest common factor) first: 6x²+9x = 3x(2x+3)
  • Recognize difference of squares instantly: x²-16=(x-4)(x+4)
  • Factor before plugging in numbers — it's usually less arithmetic
  • Canceling common factors in a fraction before multiplying avoids huge numbers

Working with Fractions Without a Calculator

Fraction fluency (common denominators, cross-multiplication) is essential when no calculator is allowed.
  • To add/subtract, find a common denominator first
  • To compare two fractions quickly, cross-multiply: a/b vs c/d compare ad vs bc
  • Convert mixed numbers to improper fractions before multiplying/dividing
  • Dividing by a fraction = multiplying by its reciprocal

Recognizing Structure (Don't Expand Everything)

The no-calculator section often rewards seeing a shortcut instead of grinding through algebra.
  • Look for a way to substitute a whole expression as one variable to simplify
  • If asked for an expression's value rather than each variable, look for a shortcut combination
  • Symmetry in answer choices is a hint you can test structure rather than solve fully
  • Plugging in small numbers (like x=1 or x=2) for variable-heavy expressions can reveal the pattern fast

Estimating Without a Calculator

Rounding cleanly lets you eliminate wrong answer choices even under no-calculator constraints.
  • Round to the nearest convenient number (10, 100, or a nice fraction)
  • Use known values: √2≈1.4, √3≈1.7, π≈3.14
  • Check whether the answer should be positive/negative and roughly what size before computing exactly
  • Eliminate answer choices that are off by a factor of 10 or have the wrong sign
Doing large multiplication digit-by-digit instead of breaking numbers into friendlier pieces (e.g., 98×6 as (100-2)×6).
Forgetting to simplify a fraction before cross-multiplying, leading to unnecessarily large numbers.
Expanding a whole expression when factoring or substitution would finish the problem in one step.
Skipping the sign check on an estimate — a negative answer choice can often be eliminated at a glance.

Overall SAT Math Timing

The Digital SAT Math section is adaptive and time-boxed — knowing your per-question budget prevents panic.
  • Digital SAT Math: 2 modules, 22 questions each, 35 minutes per module (≈1.6 min/question average)
  • Easier questions should take well under a minute; save extra time for the harder ones
  • Module 2's difficulty depends on Module 1 performance — accuracy in Module 1 matters most
  • Never leave a question blank — there's no penalty for guessing

The 'Mark and Move' Strategy

Spending too long on one hard question steals time from several easy ones you could have gotten right.
  • If stuck after ~60-90 seconds, mark the question and move on
  • Return to marked questions only after finishing the rest of the module
  • An eliminated-down-to-two guess is worth more expected points than a blank left by running out of time
  • Flagging tools in the digital SAT let you review marked items easily before submitting

Recognizing Question Difficulty Fast

Spotting a problem's real difficulty in the first few seconds helps you allocate time wisely.
  • Dense word problems often hide a simple one-step calculation — read for the actual question first
  • Multiple sub-steps (system of equations, multi-part geometry) signal a longer problem — budget more time
  • If an answer choice can be tested directly (backsolving), that's often faster than full algebra
  • Questions with clean, small numbers are usually meant to be solved quickly

Order of Operations Under Time Pressure

Choose your own order: do the fast, sure questions first regardless of their position in the module.
  • Skipping and returning is allowed — use it to build early momentum and confidence
  • Doing easy questions first banks points early even if you run out of time later
  • Don't let question order dictate your pace — a question 5 out of 22 can be harder than question 20
  • Track time at natural checkpoints (every 5-6 questions) rather than watching the clock constantly

Final-Minutes Strategy

With limited time left, maximize expected points rather than attempting to finish everything perfectly.
  • With under 2 minutes left, fill in a guess for every remaining blank question immediately
  • Prioritize questions you've partially worked through over fresh unread questions
  • For geometry/multiple-choice, eliminate any answer that's clearly the wrong sign or order of magnitude before guessing
  • Never leave the module with unanswered questions — guessing has no penalty
Spending 4-5 minutes perfecting one hard question and running out of time for three easy ones later.
Solving questions strictly in order instead of doing the fast, confident ones first.
Leaving blanks near the time limit — since there's no guessing penalty, every blank is a lost opportunity.
Not tracking time at all until it's nearly gone — check progress every 5-6 questions.
Term
Press Enter or Space to flip the card. Left and right arrows move between cards. 1 marks it known, 2 marks it still learning.
Click or press Enter to flip · Rate yourself to track weak cards
Browse all 80 flashcards as a list

Unit 1: Heart of Algebra

Slope-Intercept Form
y = mx + b, where m is the slope and b is the y-intercept.
Slope Formula
m = (y2-y1)/(x2-x1), the rise over run between two points.
System of Equations
A set of equations solved together; solution is where all are simultaneously true.
No Solution (system)
Occurs when two linear equations have the same slope but different y-intercepts (parallel lines).
Infinite Solutions (system)
Occurs when two equations represent the exact same line.
Point-Slope Form
y - y1 = m(x - x1), used when given one point and the slope.
Compound Inequality
An inequality with two bounds, e.g., -3 < 2x+1 < 7, solved across all three parts at once.
Absolute Value Equation
|x-a|=b means x is b units from a, giving two cases: x=a+b or x=a-b.
Break-Even Point
Where two cost/revenue expressions are equal — set them equal and solve.
Literal Equation
An equation with multiple variables solved for one variable in terms of the others.

Unit 2: Problem Solving & Data

Percent Change Formula
(New Value − Original Value) / Original Value × 100%.
Unit Rate
A ratio expressed per one unit, like dollars per item or miles per hour.
Mean
The sum of all values divided by the number of values; sensitive to outliers.
Median
The middle value of a data set sorted in order; resistant to outliers.
Standard Deviation
A measure of how spread out data values are from the mean.
Conditional Relative Frequency
A cell value in a two-way table divided by its row or column total.
Line of Best Fit
A line that models the overall linear trend of scatterplot data, used for predictions.
Independent Events (Probability)
Events where P(A and B) = P(A) × P(B).
Average Speed (round trip)
Total distance divided by total time — NOT the simple average of two speeds.
Scale Factor
The ratio used to enlarge or reduce a figure's dimensions proportionally.

Unit 3: Passport to Advanced Math

Quadratic Formula
x = (-b ± √(b²-4ac)) / 2a, solves any equation of the form ax²+bx+c=0.
Discriminant
b²-4ac; determines the number of real solutions to a quadratic (positive=2, zero=1, negative=0).
Difference of Squares
a² - b² = (a+b)(a-b).
Perfect Square Trinomial
a² ± 2ab + b² = (a ± b)².
Exponential Growth Model
y = a(1+r)^t, where a is the initial value and r is the growth rate.
Exponential Decay Model
y = a(1-r)^t, where a is the initial value and r is the decay rate.
Function Notation f(x)
Describes the output of a function f when the input is x.
Vertex of a Parabola
The maximum or minimum point of a quadratic graph, at x = -b/2a.
Zero Product Property
If (x-p)(x-q)=0, then x=p or x=q.
Composite Function
f(g(x)) means evaluate g first, then use that result as the input to f.

Unit 4: Geometry Essentials

Pythagorean Theorem
a² + b² = c², where c is the hypotenuse of a right triangle.
Circle Area Formula
A = πr², where r is the radius.
Circle Circumference Formula
C = 2πr, where r is the radius.
Equation of a Circle
(x-h)² + (y-k)² = r², with center (h,k) and radius r.
Volume of a Cylinder
V = πr²h, where r is the radius and h is the height.
Volume of a Sphere
V = (4/3)πr³.
Similar Triangles
Triangles with equal corresponding angles and proportional corresponding sides.
Vertical Angles
A pair of opposite angles formed by intersecting lines; always equal.
Sector Area
(central angle/360°) × πr², the area of a 'pizza slice' of a circle.
Triangle Angle Sum Theorem
The three interior angles of any triangle sum to 180°.

Unit 5: Trigonometry Basics

SOH-CAH-TOA
sin=opposite/hypotenuse, cos=adjacent/hypotenuse, tan=opposite/adjacent.
30-60-90 Triangle Ratio
Side lengths in ratio 1 : √3 : 2 (short leg : long leg : hypotenuse).
45-45-90 Triangle Ratio
Side lengths in ratio 1 : 1 : √2 (leg : leg : hypotenuse).
Pythagorean Identity
sin²θ + cos²θ = 1 for any angle θ.
Complementary Angle Identity
sin(θ) = cos(90° - θ).
Unit Circle Point
A point at angle θ on the unit circle has coordinates (cos θ, sin θ).
Radian Conversion
Multiply degrees by π/180 to convert to radians.
Inverse Trig Function
sin⁻¹, cos⁻¹, tan⁻¹ find the angle that produces a given ratio.
Hypotenuse
The side opposite the right angle; always the longest side of a right triangle.
Angle of Elevation
The angle measured upward from horizontal to a line of sight.

Unit 6: Calculator Strategies

Backsolving
Plugging in answer choices to a problem instead of solving algebraically.
1-Var Stats
A calculator function that computes mean, standard deviation, and more from a data list.
Graphing Calculator Intersect Feature
Finds where two graphed equations cross, solving a system visually.
Estimation Check
Rounding numbers to sanity-check a calculator's exact output before trusting it.
Table Feature
Generates x/y value pairs for a function, useful for testing multiple inputs quickly.
LinReg (Linear Regression)
A calculator tool that finds the line of best fit for a data set.
Order of Operations (Calculator Entry)
Use extra parentheses to group full numerators/denominators when typing fractions.
Standard Deviation (σx)
Calculator output measuring the spread of a data set from its mean.
Answer Elimination
Ruling out choices that are clearly the wrong sign or order of magnitude before computing.
Regression Correlation (r)
A calculator statistic showing how well a line fits scatterplot data, from -1 to 1.

Unit 7: No-Calculator Tactics

Mental Math Decomposition
Breaking a number into friendlier parts, e.g., 98×6 = (100-2)×6.
Greatest Common Factor (GCF)
The largest factor shared by all terms, pulled out first when factoring.
√2 Approximation
√2 ≈ 1.41, useful for quick no-calculator estimates.
√3 Approximation
√3 ≈ 1.73, useful for quick no-calculator estimates.
π Approximation
π ≈ 3.14, used for no-calculator circle estimates.
Reciprocal
The reciprocal of a/b is b/a; dividing by a fraction means multiplying by its reciprocal.
Cross-Multiplication
For a/b vs c/d, compare ad vs bc to determine which fraction is larger without a calculator.
Common Fraction-Percent Pairs
1/2=50%, 1/4=25%, 1/5=20%, 1/8=12.5%, 3/4=75%.
Difference of Squares Shortcut
x² - n² factors instantly as (x-n)(x+n).
Substitution Shortcut
Replacing a repeated expression with a single variable to simplify complex algebra fast.

Unit 8: Timing & Pacing Drills

Digital SAT Math Structure
2 modules of 22 questions each, 35 minutes per module.
Average Pace
About 1.6 minutes per question on the Digital SAT Math section.
Mark-and-Move
Flagging a hard question to return to later instead of stalling on it.
Adaptive Testing
Module 2's difficulty is set based on how well you performed on Module 1.
No Guessing Penalty
There is no point deduction for a wrong answer, so never leave a question blank.
Checkpoint Pacing
Checking your time every 5-6 questions rather than constantly watching the clock.
Expected Value of Guessing
Even a random guess has positive expected value when there's no penalty for wrong answers.
Question Order Flexibility
You may answer questions in any order within a module — use it to bank easy points first.
Time Budget per Question
Roughly 90 seconds is a reasonable cutoff before marking a question and moving on.
Final-Minutes Rule
Fill in a guess for every remaining blank as time runs out — something beats nothing.
Press 1–4 to answer · Enter for next

Unit 1: Heart of Algebra

Linear Equations in One Variable
Isolate the variable using inverse operations, applying the same operation to both sides to keep the equation balanced.
Systems of Linear Equations
A system's solution is the point (or points) where all equations are simultaneously true — solve by substitution, elimination, or graphing.
Linear Word Problems & Modeling
Translate real-world scenarios into equations: identify the rate (slope) and starting value (y-intercept).
Linear Inequalities
Inequalities behave like equations except the sign flips when multiplying or dividing by a negative number.
Interpreting Linear Graphs
The slope and intercept of a line carry real-world meaning that the SAT tests directly.
Absolute Value Equations
|x| represents distance from 0, so absolute value equations split into two cases.
Key fact
Slope-intercept form: y = mx + b (m = slope, b = y-intercept).
Key fact
A system of two linear equations has 0, 1, or infinitely many solutions — never exactly 2.
Key fact
To solve by elimination, make one variable's coefficients opposites, then add the equations.
Key fact
Point-slope form: y - y1 = m(x - x1), useful when given a point and a slope.

Unit 2: Problem Solving & Data

Ratios, Rates, and Proportions
A ratio compares two quantities; a proportion sets two ratios equal and is solved by cross-multiplication.
Percentages
Percent means 'per hundred'; convert to a decimal (divide by 100) before computing.
Mean, Median, and Standard Deviation
Measures of center describe a 'typical' value; spread measures describe how varied the data is.
Reading Tables, Scatterplots & Two-Way Tables
Extract exact values from tables and describe trends/associations from scatterplots.
Unit Conversion & Rates
Use dimensional analysis: multiply by conversion fractions so unwanted units cancel.
Probability
Probability = favorable outcomes ÷ total possible outcomes, always between 0 and 1.
Key fact
Percent change = (new value − original value) / original value × 100%.
Key fact
Mean is sensitive to outliers; median is not.
Key fact
Distance = Rate × Time; average speed for a round trip = total distance ÷ total time.
Key fact
In a two-way table, conditional relative frequency divides by the relevant row or column total, not the grand total.

Unit 3: Passport to Advanced Math

Factoring Quadratics
Rewrite ax²+bx+c as a product of binomials to reveal roots quickly.
The Quadratic Formula & Discriminant
x = (-b ± √(b²-4ac)) / 2a solves any quadratic ax²+bx+c=0.
Exponent Rules
Exponent rules let you simplify expressions without expanding them.
Exponential Growth & Decay
Exponential functions model repeated percent change: y = a(1 ± r)^t.
Function Notation & Transformations
f(x) notation describes input-output relationships; shifting the graph changes the equation predictably.
Systems with a Quadratic and a Line
A line and a parabola can intersect at 0, 1, or 2 points — set the equations equal to find them.
Key fact
Quadratic formula: x = (-b ± √(b²-4ac)) / (2a).
Key fact
Difference of squares: a² - b² = (a+b)(a-b).
Key fact
Exponential model: y = a(1 ± r)^t, where a is the initial value and r is the rate.
Key fact
Discriminant b²-4ac determines the number of real solutions: positive→2, zero→1, negative→0.

Unit 4: Geometry Essentials

Area and Perimeter
Know the standard area/perimeter formulas for common shapes cold — the SAT rarely gives them for basic figures.
The Pythagorean Theorem
In any right triangle, a² + b² = c², where c is the hypotenuse (the side opposite the right angle).
Circles: Arcs, Sectors, and Angles
A circle has 360° total; arcs and sectors are fractional parts of the whole circle.
Volume and Surface Area (3D Solids)
3D formulas extend 2D area formulas by adding a height/depth dimension.
Similar Triangles & Congruence
Similar figures have proportional corresponding sides and equal corresponding angles.
Angle Relationships
Parallel lines cut by a transversal create predictable, testable angle pairs.
Key fact
Pythagorean theorem: a² + b² = c² (c = hypotenuse).
Key fact
Circle area = πr²; circumference = 2πr; equation of a circle: (x-h)² + (y-k)² = r².
Key fact
Sum of interior angles of a triangle = 180°; of a quadrilateral = 360°.
Key fact
If similar figures have scale factor k, areas scale by k² and volumes scale by k³.

Unit 5: Trigonometry Basics

SOH-CAH-TOA
In a right triangle, trig ratios relate an acute angle to the ratio of two sides.
Special Right Triangles
The 45-45-90 and 30-60-90 triangles have fixed, memorizable side ratios.
Complementary Angles: sin/cos Relationship
In a right triangle, the two acute angles are complementary (sum to 90°), which links sine and cosine directly.
The Unit Circle (Basics)
The unit circle extends trig ratios to all angles using a circle of radius 1 centered at the origin.
Solving for Missing Sides/Angles with Trig
Use trig ratios and inverse trig functions to find unknown sides or angles in right triangles.
Key fact
SOH-CAH-TOA: sin=opp/hyp, cos=adj/hyp, tan=opp/adj.
Key fact
30-60-90 triangle side ratio: 1 : √3 : 2. 45-45-90 triangle side ratio: 1 : 1 : √2.
Key fact
Pythagorean identity: sin²θ + cos²θ = 1.
Key fact
sin(θ) = cos(90° − θ) for complementary angles in a right triangle.

Unit 6: Calculator Strategies

When (and When Not) to Use Your Calculator
The calculator section allows a calculator, but the fastest solution is often mental math or estimation, not typing every step.
Solving Systems Graphically
A graphing calculator finds intersection points instantly instead of solving algebraically.
Using Statistical Functions
Calculators can compute mean, median, and standard deviation directly from a list of data.
Table Feature for Function Evaluation
The table (TABLE) feature evaluates a function at many x-values at once — faster than plugging in one at a time by hand.
Answer-Choice Elimination (Backsolving)
On calculator-allowed problems, plugging answer choices back into the problem is often faster than solving algebraically.
Key fact
Estimating an answer's rough size BEFORE calculating catches typing errors instantly.
Key fact
1-Var Stats (or equivalent) gives mean and standard deviation directly from an entered data list.
Key fact
Backsolving (plugging in answer choices) is often faster than algebra on calculator-section problems with ugly numbers.
Key fact
A graphing calculator's 'intersect' function solves systems of equations, including nonlinear ones, without algebra.

Unit 7: No-Calculator Tactics

Mental Math Shortcuts
The no-calculator section rewards quick, exact mental arithmetic and smart simplification over brute-force computation.
Factoring to Simplify Fast
Factoring often turns an intimidating no-calculator expression into a one-step answer.
Working with Fractions Without a Calculator
Fraction fluency (common denominators, cross-multiplication) is essential when no calculator is allowed.
Recognizing Structure (Don't Expand Everything)
The no-calculator section often rewards seeing a shortcut instead of grinding through algebra.
Estimating Without a Calculator
Rounding cleanly lets you eliminate wrong answer choices even under no-calculator constraints.
Key fact
√2 ≈ 1.41, √3 ≈ 1.73, π ≈ 3.14 — memorize these for quick no-calculator estimates.
Key fact
GCF factoring first (e.g., 6x²+9x = 3x(2x+3)) usually beats expanding and re-factoring.
Key fact
Common fraction–decimal–percent equivalents: 1/2=50%, 1/4=25%, 1/5=20%, 1/8=12.5%, 3/4=75%.
Key fact
Dividing by a fraction means multiplying by its reciprocal: a ÷ (b/c) = a × (c/b).

Unit 8: Timing & Pacing Drills

Overall SAT Math Timing
The Digital SAT Math section is adaptive and time-boxed — knowing your per-question budget prevents panic.
The 'Mark and Move' Strategy
Spending too long on one hard question steals time from several easy ones you could have gotten right.
Recognizing Question Difficulty Fast
Spotting a problem's real difficulty in the first few seconds helps you allocate time wisely.
Order of Operations Under Time Pressure
Choose your own order: do the fast, sure questions first regardless of their position in the module.
Final-Minutes Strategy
With limited time left, maximize expected points rather than attempting to finish everything perfectly.
Key fact
Digital SAT Math: 2 modules × 22 questions, 35 minutes each (~1.6 minutes per question on average).
Key fact
There is no penalty for guessing — never leave a question blank.
Key fact
Module 2 difficulty is determined by your Module 1 performance, so early accuracy matters most.
Key fact
Mark-and-move: if stuck 60-90 seconds, flag the question and return to it later.
Common mistakes for each unit — read the mistake, then make sure you know why it's wrong.

Unit 1: Heart of Algebra

Watch out
Forgetting to flip the inequality sign when multiplying/dividing by a negative — always flip it, not just when the answer 'looks wrong.'
Watch out
Distributing incorrectly: -2(x - 3) = -2x + 6, not -2x - 6.
Watch out
Answering with the value of the wrong variable — the question may ask for y after you solved for x.
Watch out
Assuming parallel lines intersect somewhere; same slope + different intercept = no solution, always.

Unit 2: Problem Solving & Data

Watch out
Using the wrong 'original' value in a percent change problem — always divide by the STARTING value, not the ending one.
Watch out
Averaging two speeds directly for a round trip instead of using total distance ÷ total time.
Watch out
Confusing mean and median — outlier-heavy data shifts the mean, not necessarily the median.
Watch out
Forgetting that successive percent changes compound rather than add (a 10% increase then 10% decrease is NOT back to the original).

Unit 3: Passport to Advanced Math

Watch out
Forgetting the ± in the quadratic formula — a quadratic almost always has two roots, not one.
Watch out
Mixing up growth and decay: (1+r) grows, (1-r) decays — using the wrong sign inverts the whole model.
Watch out
Sign errors when factoring: (x-3)(x+3) = x²-9, not x²+9.
Watch out
Applying exponent rules to terms that are ADDED, not multiplied — x^a + x^b does NOT simplify to x^(a+b).

Unit 4: Geometry Essentials

Watch out
Using the diameter instead of the radius in area/circumference formulas — always confirm which one is given.
Watch out
Forgetting that similar triangles scale AREA by k², not by k — a doubled side length means 4× the area.
Watch out
Mislabeling the hypotenuse — it's always the side opposite the right angle, and the longest side.
Watch out
Confusing arc length (a length, uses 2πr) with sector area (a region, uses πr²).

Unit 5: Trigonometry Basics

Watch out
Mixing up which side is 'opposite' vs. 'adjacent' — it depends entirely on which angle you're referencing.
Watch out
Using degree-mode formulas with a calculator set to radians (or vice versa) — always check the mode.
Watch out
Forgetting that the hypotenuse is always the LONGEST side and is never labeled 'opposite' or 'adjacent.'
Watch out
Assuming SOH-CAH-TOA works on non-right triangles — it only applies to right triangles.

Unit 6: Calculator Strategies

Watch out
Blindly trusting calculator output without an estimate to sanity-check — a single mistyped digit can produce a wildly wrong (but confident-looking) answer.
Watch out
Re-typing a whole expression instead of editing after a typo, wasting valuable time.
Watch out
Using calculator mode inconsistently (degrees vs. radians) for trig problems.
Watch out
Forgetting order of operations when typing complex fractions — use extra parentheses around the entire numerator and denominator.

Unit 7: No-Calculator Tactics

Watch out
Doing large multiplication digit-by-digit instead of breaking numbers into friendlier pieces (e.g., 98×6 as (100-2)×6).
Watch out
Forgetting to simplify a fraction before cross-multiplying, leading to unnecessarily large numbers.
Watch out
Expanding a whole expression when factoring or substitution would finish the problem in one step.
Watch out
Skipping the sign check on an estimate — a negative answer choice can often be eliminated at a glance.

Unit 8: Timing & Pacing Drills

Watch out
Spending 4-5 minutes perfecting one hard question and running out of time for three easy ones later.
Watch out
Solving questions strictly in order instead of doing the fast, confident ones first.
Watch out
Leaving blanks near the time limit — since there's no guessing penalty, every blank is a lost opportunity.
Watch out
Not tracking time at all until it's nearly gone — check progress every 5-6 questions.