SAT Math

How the digital SAT actually works, the tools you get, and a diagnostic that decides where you start.

• · How the digital SAT works: 2 sections, 4 modules, adaptive routing, timing
• · What's on Math: the four domains and their weights
• · The test-day interface: tools, mark for review, eliminator, navigator
• · Grid-in rules — formatting, fractions, decimals, negatives
• · Drill It
• · Desmos basics: entering expressions, graphing, zoom, the table
• · The reference sheet: what's given, what you must memorise
• · Diagnostic (Math) — sets your starting module

The arithmetic and algebra fluency everything later assumes. Skippable via the diagnostic.

• · Fractions, decimals, percents — conversion and fluency
• · Drill It
• · Order of operations, negatives, absolute value
• · Drill It
• · Exponent rules: product, quotient, power, zero, negative
• · Drill It
• · Radicals and rational exponents; simplifying; √(x²) = |x|
• · Drill It
• · Translating English into algebra: "of", "per", "less than", "is"
• · Drill It
• · Units and dimensional analysis
• · Drill It
• · Module cheat sheet
• · Module quiz

Solving, the no-solution and infinite-solution cases, and the sign-flip rule.

• · Solving linear equations; clearing fractions
• · Drill It
• · No solution / infinitely many solutions — and how items force each case
• · Drill It
• · Absolute value equations: two cases, and when there are 0 or 1
• · Drill It
• · Linear inequalities; the sign-flip rule
• · Drill It
• · Compound inequalities and "at least / at most" translation
• · Drill It
• · Worked example: solving for one variable in terms of others
• · Module cheat sheet
• · Module quiz

Slope as a rate, the three forms, and what a line means in context.

• · Slope as rate of change; the slope formula
• · Drill It
• · Slope-intercept, point-slope, standard form — converting between them
• · Drill It
• · Reading a line: intercepts, what slope and intercept mean in context
• · Drill It
• · Parallel and perpendicular slopes
• · Drill It
• · Horizontal and vertical lines; undefined slope
• · Drill It
• · Desmos: graphing a line, finding intercepts, using sliders
• · Module cheat sheet
• · Module quiz

Substitution, elimination, and the proportionality test for how many solutions exist.

• · Substitution
• · Drill It
• · Elimination
• · Drill It
• · Number of solutions: the a₁/a₂ = b₁/b₂ ≠ c₁/c₂ test — one / none / infinite
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · Systems from word problems: the count equation and the value equation
• · Drill It
• · Desmos: solving systems by intersection
• · Timed mixed set
• · Module cheat sheet
• · Module quiz

Turning a described situation into a model, and reading the coefficients back out.

• · Building a linear model: the fixed part and the per-unit part
• · Drill It
• · Interpreting coefficients in context, with units
• · Drill It
• · Systems of inequalities as constraints; feasible regions
• · Drill It
• · Worked example: a multi-constraint real-world item
• · Module cheat sheet
• · Module quiz

Proportional reasoning, and why rates cannot simply be averaged.

• · Setting up proportions correctly
• · Drill It
• · Unit rates; combined rates and why you cannot average rates
• · Drill It
• · Work and distance-rate-time problems
• · Drill It
• · Unit conversion by cancelling units
• · Drill It
• · Similar figures: the k, k², k³ scaling rule
• · Drill It
• · Module cheat sheet
• · Module quiz

The multiplier method, and the successive-change trap the test relies on.

• · Percent of, percent change, percent more or less than
• · Drill It
• · The multiplier method: ×(1 ± r)
• · Drill It
• · Trap card: successive percent changes multiply — 20% up then 20% down is not back to start
• · Drill It
• · Reverse percentages: finding the original
• · Drill It
• · Module cheat sheet
• · Module quiz

Centre, spread and shape — and which measure a question is actually asking for.

• · Mean, median, mode, range
• · Drill It
• · Outliers: what moves the mean and what does not
• · Drill It
• · Weighted averages; finding a missing value from the mean
• · Drill It
• · Reading dot plots, histograms and frequency tables — weight by frequency
• · Drill It
• · Box plots, quartiles, IQR
• · Drill It
• · Standard deviation on the SAT is conceptual: compare spread, never compute
• · Drill It
• · Skew: mean vs median in right-, left- and symmetric distributions
• · Drill It
• · Module cheat sheet
• · Module quiz

Scatterplots, lines of best fit, residuals, and choosing the model shape.

• · Scatterplots and association
• · Drill It
• · Line of best fit: slope and intercept in context; predicting
• · Drill It
• · Residuals; over- and under-prediction
• · Drill It
• · Choosing the model shape: linear vs exponential vs quadratic
• · Drill It
• · Correlation is not causation
• · Drill It
• · Desmos: regressions (y₁ ~ mx₁ + b)
• · Module cheat sheet
• · Module quiz

Finding the right denominator — which is the whole difficulty of this domain.

• · Basic probability; complements
• · Drill It
• · Two-way tables: finding the right total
• · Drill It
• · Conditional probability — "given that" changes the denominator
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · Independent and mutually exclusive events
• · Drill It
• · Expected counts from a proportion
• · Drill It
• · Module cheat sheet
• · Module quiz

What a sample licenses you to claim, and what it does not.

• · Sample vs population; when you can generalise
• · Drill It
• · Margin of error: interpreting the interval, sample size, confidence level
• · Drill It
• · Random selection vs random assignment — the 2×2 grid of valid conclusions
• · Drill It
• · Evaluating whether a stated conclusion is supported
• · Drill It
• · Module cheat sheet
• · Module quiz

Notation, composition, domain and range, and the inside-opposite transformation rule.

• · Function notation; evaluating f(a) and f(a+2)
• · Drill It
• · Solving f(x) = k; reading solutions off a graph
• · Drill It
• · Composition f(g(x))
• · Drill It
• · Domain and range, including real-world restrictions
• · Drill It
• · Transformations: shifts, reflections, stretches
• · Drill It
• · Trap card: f(x + h) moves LEFT — the inside/opposite rule
• · Drill It
• · Absolute value functions
• · Drill It
• · Module cheat sheet
• · Module quiz

Three forms, what each one hands you for free, and four ways to solve.

• · Standard vs factored vs vertex form — what each one tells you
• · Drill It
• · Factoring: GCF, x² + bx + c, difference of squares, perfect squares
• · Drill It
• · Factoring when a ≠ 1 (AC method, grouping)
• · Drill It
• · The quadratic formula
• · Drill It
• · Completing the square
• · Drill It
• · Worked example: choosing the fastest solving method
• · Module cheat sheet
• · Module quiz

Where the parabola turns, how many times it crosses, and what the roots sum to.

• · Finding the vertex and min/max: h = −b/(2a), k = f(h); vertex form by inspection
• · Drill It
• · Rule card: the maximum VALUE is k; the x that produces it is h — read the question
• · Drill It
• · Axis of symmetry; direction of opening from a
• · Drill It
• · Number of solutions: the discriminant b² − 4ac
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · Sum and product of roots (−b/a, c/a)
• · Drill It
• · Quadratic word problems: projectile height, area, revenue
• · Drill It
• · Desmos: vertex, zeros, and finding k for exactly one solution with a slider
• · Timed mixed set
• · Module cheat sheet
• · Module quiz

The factor theorem's four equivalent statements, and reading a graph from its factors.

• · Operations and expansion
• · Drill It
• · The factor theorem: zero ⟺ factor ⟺ x-intercept ⟺ P(k) = 0
• · Drill It
• · The remainder theorem
• · Drill It
• · Zeros, multiplicity, and whether the graph crosses or touches
• · Drill It
• · End behaviour from degree and leading coefficient
• · Drill It
• · Module cheat sheet
• · Module quiz

The growth factor, percent change over time, and telling linear from exponential.

• · y = a·bˣ: initial value and growth factor
• · Drill It
• · Percent growth and decay: a(1 ± r)ᵗ
• · Drill It
• · Compounding and other periods: a(1 + r/n)^(nt)
• · Drill It
• · Half-life and doubling time
• · Drill It
• · Linear vs exponential from a table: constant difference vs constant ratio
• · Drill It
• · Module cheat sheet
• · Module quiz

Simplifying by factoring, the LCD, and why every answer must be checked.

• · Simplifying by factoring; excluded values
• · Drill It
• · Adding, subtracting, multiplying, dividing
• · Drill It
• · Solving rational equations with the LCD
• · Drill It
• · Extraneous solutions — always check
• · Drill It
• · Asymptotes and holes, conceptually
• · Drill It
• · Module cheat sheet
• · Module quiz

Squaring both sides, and counting intersections with the discriminant.

• · Solving radical equations; squaring both sides
• · Drill It
• · Extraneous solutions again — the SAT tests this on purpose
• · Drill It
• · Line ∩ parabola: substitute, then use the discriminant to count intersections
• · Drill It
• · Circle ∩ line
• · Drill It
• · Desmos: counting intersections visually
• · Module cheat sheet
• · Module quiz

The angle facts, similarity, and the triangles worth memorising.

• · Supplementary, complementary and vertical angles
• · Drill It
• · Parallel lines and a transversal
• · Drill It
• · Triangle angle sum; exterior angle; triangle inequality
• · Drill It
• · Isosceles and equilateral properties
• · Drill It
• · Similar triangles and setting up the right proportion
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · Congruence criteria
• · Drill It
• · Pythagorean theorem and the common triples
• · Drill It
• · Special right triangles: 45-45-90 and 30-60-90
• · Drill It
• · Module cheat sheet
• · Module quiz

The circle equation, inscribed angles, and radians.

• · Interior and exterior angle sums; regular polygons
• · Drill It
• · Quadrilateral properties, perimeter, area
• · Drill It
• · Circle basics: area, circumference
• · Drill It
• · The circle equation (x−h)² + (y−k)² = r², and completing the square to get there
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · Central vs inscribed angles; the semicircle right angle
• · Drill It
• · Arc length and sector area, in degrees and radians
• · Drill It
• · Radians: conversion and θ = s/r
• · Drill It
• · Tangent ⟂ radius
• · Drill It
• · Module cheat sheet
• · Module quiz

Volume, the scaling rule, and proving shapes with slope and distance.

• · Volume and surface area; composite solids
• · Drill It
• · Scaling: k, k², k³
• · Drill It
• · Distance and midpoint
• · Drill It
• · Finding an endpoint from a midpoint; proving shapes with slope and distance
• · Drill It
• · Module cheat sheet
• · Module quiz

SOH-CAH-TOA, the complementary identity, and what is not tested.

• · SOH-CAH-TOA relative to the referenced angle
• · Drill It
• · sin θ = cos(90° − θ) — the complementary identity
• · Drill It (1 of 2)
• · Drill It (2 of 2)
• · sin²θ + cos²θ = 1 and tan θ = sin θ / cos θ
• · Drill It
• · Trig with special right triangles
• · Drill It
• · Scope card: what is NOT tested — unit circle, law of sines/cosines, inverse trig, calculus
• · Module cheat sheet
• · Module quiz

Every technique worth knowing, and the items where reaching for it costs you time.

• · Solving any equation by graphing both sides
• · Drill It
• · Systems and intersections at speed
• · Drill It
• · Zeros, vertex, max and min
• · Drill It
• · Sliders for "find the value of k such that…"
• · Drill It
• · Tables, sequences and data entry
• · Drill It
• · Regressions for line-of-best-fit items
• · Drill It
• · Inequalities and shaded regions
• · Drill It
• · When NOT to use Desmos — conceptual, symbolic-answer and trivial-arithmetic items
• · Timed set: 12 items, Desmos-optimal vs algebra-optimal side by side

Backsolving, picking numbers, and answering the question that was actually asked.

• · Backsolving from the answer choices
• · Drill It
• · Picking numbers for variable-in-the-choices items
• · Drill It
• · Answer the right question: they asked for 2x, not x
• · Drill It
• · Grid-in formatting under time; character limits; repeating decimals
• · Drill It
• · Pacing: about 95 s per question, when to mark and move, guessing policy
• · Pacing set
• · Module 1 strategy and what adaptive routing means for you

Simulations at real pace, and how to review one so it changes your next score.

• · Math Module 1 simulation (22 q / 35 min)
• · Math Module 2 — lower tier
• · Math Module 2 — upper tier (the hard-module drill)
• · How to review a practice test properly: the error log method
• · Building your personal weakness queue from a score report