Test format & strategy
The Digital SAT, taken on a computer, is section-adaptive: how you do on the first math module sets the difficulty of the second. The good news for math — a graphing calculator is built in and allowed throughout.
Math runs as two adaptive modules (about 44 questions in roughly 70 minutes). Questions are either multiple choice or student-produced responses (grid-ins). The on-screen Desmos graphing calculator is available on every question, and the math score is reported on the 200–800 scale.
📊 Digital SAT Math — Exam Blueprint
Source: College Board — SAT Suite of Assessments Math Test Specification
Graph it in Desmos the moment you're stuck; plug in the answer choices or convenient numbers; and on grid-ins, don't round early — enter a fraction or a full decimal.
Linear Equations & Systems
📌 Domain: Algebra · Linear equations in one variable · Linear inequalities in one variable · Systems of two linear equations in two variables
Linear relationships are the backbone of the algebra domain: solving for a variable, reading slope and intercept, and handling two equations at once.
Slope-intercept form is \( y=mx+b \); slope \( =\dfrac{y_2-y_1}{x_2-x_1} \). Solve a system by substitution or elimination — or graph both lines in Desmos and read the intersection.
Solve \( 3x+5=20 \).
Solve the system \( x+y=10 \), \( x-y=2 \). Add the equations: \( 2x=12 \).
Linear Functions & Inequalities
📌 Domain: Algebra · Linear functions · Linear inequalities in one or two variables
A linear function models a constant rate of change; an inequality describes a region rather than a single line.
For \( f(x)=mx+b \), \( m \) is the rate and \( b \) the starting value. Parallel lines share a slope; perpendicular slopes multiply to \( -1 \). An inequality's boundary line is dashed for \( < \) or \( > \), solid for \( \le \) or \( \ge \).
Adjust the slope \( m \) and intercept \( b \) of \( y=mx+b \). The amber dots mark the intercepts the SAT most often asks for.
| x | y |
|---|
The y-intercept is \( (0,b) \); the x-intercept solves \( mx+b=0 \).
If \( f(x)=3x+1 \), find \( f(2)-f(0) \).
Find the x-intercept of \( 2x+4y=8 \). Set \( y=0 \): \( 2x=8 \).
Quadratics & Nonlinear Functions
📌 Domain: Advanced Math · Nonlinear equations in one variable · Nonlinear functions · Systems of equations in two variables
The advanced-math domain leans hard on quadratics: factoring, the quadratic formula, the vertex, and what the discriminant reveals.
\( x=\dfrac{-b\pm\sqrt{b^{2}-4ac}}{2a} \); the discriminant \( b^{2}-4ac \) counts real solutions. The vertex of \( y=ax^{2}+bx+c \) is at \( x=-\dfrac{b}{2a} \); factored form \( a(x-r_1)(x-r_2) \) shows the roots directly.
Solve \( x^{2}-5x+6=0 \). \( (x-2)(x-3)=0 \):
State the vertex of \( y=(x-3)^{2}+2 \).
Solve \(|ax+b|=c\) by splitting into two linear equations: \(ax+b=c\) and \(ax+b=-c\). Always check both. For inequalities: \(|x|<k\;\Rightarrow\;-k<x<k\) (and-compound); \(|x|>k\;\Rightarrow\;x<-k\;\text{or}\;x>k\) (or-compound). No solution if right side is negative.
Solve \(|2x-3|=7\). Case 1: \(2x-3=7\Rightarrow x=5\). Case 2: \(2x-3=-7\Rightarrow x=-2\). Both check out. Solutions: \(x=5\) or \(x=-2\).
A linear–quadratic system can have 0, 1, or 2 solutions — determined by the discriminant after substitution. Substitute the linear expression for \(y\) into the quadratic, rearrange to standard form, and solve or test the discriminant. For a line and circle, the same method applies.
Find all intersections of \(y=x^{2}\) and \(y=x+2\). Set equal: \(x^{2}=x+2\Rightarrow x^{2}-x-2=0\Rightarrow(x-2)(x+1)=0\).
Exponentials, Polynomials & Rational Expressions
📌 Domain: Advanced Math · Equivalent expressions · Nonlinear functions · Nonlinear equations (radical, rational)
Beyond quadratics: exponential growth and decay, exponent rules, and simplifying polynomial and rational expressions.
\( a^{m}a^{n}=a^{m+n} \) and \( (a^{m})^{n}=a^{mn} \). An exponential model \( y=a\cdot b^{x} \) grows when \( b>1 \) and decays when \( 0<b<1 \); \( a \) is the starting value.
In \( y=2\cdot b^{x} \), a base \( b>1 \) grows while \( b<1 \) decays. Slide \( b \) and watch the curve flip behaviour around \( b=1 \).
| x | y |
|---|
Each \( +1 \) in \( x \) multiplies \( y \) by \( b \); the start value \( (0,2) \) is amber.
Solve \( 2^{x}=16 \).
Simplify \( \dfrac{6x^{2}}{2x} \).
Isolate the radical, then square both sides. Always check your answers — squaring can introduce extraneous solutions that don't satisfy the original equation. Require the expression under the radical to be \(\geq0\) (domain restriction).
Solve \(\sqrt{3x+1}=4\). Square: \(3x+1=16\Rightarrow x=5\). Check: \(\sqrt{16}=4\). \checkmark
Multiply through by the LCD to clear fractions. Identify domain restrictions (values making any denominator zero) before solving. Discard any solution that equals a restricted value.
Solve \(\dfrac{3}{x-1}+\dfrac{1}{x+1}=\dfrac{4}{x^{2}-1}\). Note \(x\neq\pm1\). Multiply by \((x-1)(x+1)\): \(3(x+1)+(x-1)=4\Rightarrow4x+2=4\Rightarrow x=\tfrac{1}{2}\). \checkmark
Check \(x=3\): \(\sqrt{1}\neq-1\). Extraneous. Check \(x=7\): \(\sqrt{9}=3\). \checkmark
One valid solution: \(x=7\).
Problem-Solving & Data Analysis
📌 Domain: Problem-Solving & Data Analysis · Ratios, rates, proportional relationships · Percentages · One-variable data (center, spread) · Two-variable data · Probability & conditional probability
The most word-heavy domain: ratios, rates and percentages, units, statistics, probability, and reading models from data.
Percent of: \( \tfrac{p}{100}\times N \); a unit rate divides to “per one”. Mean is total over count; the line of best fit models a linear trend in a scatterplot.
Slide the trend line's slope to shrink the total squared residual (SSR). The smallest SSR marks the line of best fit.
| x | actual y | predicted y | residual | residual² |
|---|
The line pivots through the data's centre; the best-fit slope here is about \( 0.95 \).
Find \( 20\% \) of \( 150 \).
A $40 shirt is marked down \( 30\% \). Find the sale price.
Mean (average) is sensitive to outliers; median (middle value, sorted) is resistant. Range = max − min. IQR = Q3 − Q1 (middle 50 % of data; also resistant to outliers). Standard deviation: the SAT never asks you to compute it — it asks you to compare or interpret it. A larger SD means greater spread around the mean.
Data: 3, 7, 8, 12, 14, 18, 21. Median = 4th value = 12. Q1 = median of {3,7,8} = 7. Q3 = median of {14,18,21} = 18.
A random sample allows generalisation to the population it was drawn from — no further. Voluntary-response and convenience samples are biased. Observational studies can show association only; establishing causation requires a randomised controlled experiment (random assignment to groups). A margin of error creates a confidence interval: estimate ± margin of error.
A poll of 600 voters finds 54 % support Candidate A, margin of error ±3 %. True support is estimated to be between 51 % and 57 %. Since the entire interval exceeds 50 %, the data support a majority — but only for the population sampled.
(A) Tutoring causes higher scores.
(B) There is an association but causation cannot be determined.
(C) Results apply to all students nationwide.
(D) The sample size is too small to draw any conclusion.
Geometry & Trigonometry
📌 Domain: Geometry & Trigonometry · Area and volume · Lines, angles, and triangles · Right triangles and trigonometry · Circles
The smallest domain by count, but reliable points: area and volume, angle relationships, right triangles and basic trigonometry, and circles.
Triangle angles sum to \( 180^{\circ} \); Pythagoras \( a^{2}+b^{2}=c^{2} \). Circle area \( \pi r^{2} \), circumference \( 2\pi r \). Right-triangle ratios: SOH-CAH-TOA.
A right triangle has legs \( 6 \) and \( 8 \). Find the hypotenuse.
Find the area of a circle with radius \( 3 \).
These are given in the SAT reference sheet — know what each variable represents so you can apply them quickly.
A cylinder has radius 3 and height 5.
When a transversal cuts two parallel lines: alternate interior angles are equal; corresponding angles are equal; co-interior (same-side interior) angles are supplementary (sum to 180°). Vertical angles are always equal regardless of parallel lines.
Two parallel lines are cut by a transversal. One angle is 65°. Find its co-interior angle.
Both are simply a fraction of the whole circle, where the fraction is \(\theta/360\).
Circle of radius 6, central angle 120°.
Calculator & Strategy
📌 All domains · Calculator strategy · Process of elimination · Time management
The built-in Desmos graphing calculator is the Digital SAT's secret weapon. Many “solve” and “system” questions become a glance at a graph.
Type an equation to graph it instantly; click a curve to read its intercepts; graph two equations and click the intersection to solve a system; build a table for data; and add a slider for “for what value of \( k \)” questions.
Student-produced responses accept fractions or decimals — enter \( \tfrac{3}{8} \) or \( .375 \), and avoid rounding unless the question tells you to. If several answers are valid, any one is accepted.
Functions & Function Notation
📌 Domain: Advanced Math · Nonlinear functions · Equivalent expressions · Piecewise and absolute value functions
Evaluating, interpreting and transforming functions runs through the Advanced Math domain.
Evaluate \( f(a) \) by substitution and read features (intercepts, vertex) from graphs. \( f(x)+k \) shifts the graph vertically; \( f(x-h) \) shifts it horizontally.
If \( f(x)=x^{2}-1 \), find \( f(4) \).
A piecewise function uses a different rule on each part of its domain. To evaluate, find which condition \(x\) satisfies, then apply only that piece. The absolute value function is itself piecewise: \(f(x)=|x|=x\) when \(x\geq0\) and \(-x\) when \(x<0\). Its graph is a V-shape with vertex where the expression equals zero.
\(f(x)=\begin{cases}x^{2} & x<2 \\ 3x-1 & x\geq2\end{cases}\). Find \(f(-1)\) and \(f(4)\). \(f(-1)\): use \(x^2\). \((-1)^2=\mathbf{1}\). \(f(4)\): use \(3x-1\). \(3(4)-1=\mathbf{11}\).
Statistics, Sampling & Margin of Error
📌 Domain: Problem-Solving & Data Analysis · One-variable data (distributions, center, spread) · Inference from sample statistics, confidence intervals, and margin of error · Evaluating statistical claims
Around 4–6 Digital SAT questions target statistics: reading distributions, choosing the right measure of center or spread, and interpreting survey results with a stated margin of error.
Mean (average) — pulled toward outliers. Median — middle value; resistant to outliers. Standard deviation — typical distance from the mean; the SAT asks you to compare or interpret SD, never to compute it. A histogram skewed right has mean > median; skewed left has mean < median.
Eight salaries (thousands): 30, 32, 35, 36, 37, 38, 40, 120. Mean \(=368/8=46\). Median \(=(36+37)/2=36.5\). The outlier (120) inflates the mean. The median better represents a typical salary.
Set A: {10,10,10,10}. Set B: {4,8,12,16}. Both have mean 10. Set A has SD = 0 (no spread). Set B has a larger SD because values vary around the mean.
A poll of 600 voters gives 54 % support with margin of error ±3 %. The true population support is estimated between 51 % and 57 %. Since the entire interval exceeds 50 %, the data suggest majority support — but only for the population the sample represents.
(A) Set B values are closer to the mean.
(B) Set B has greater variability.
(C) The two sets have the same range.
(D) Set A has more data points.
(A) Exactly 60 % of all students prefer longer lunches.
(B) Between 55 % and 65 % of all students at the school prefer longer lunches.
(C) Results apply to all students nationwide.
(D) No conclusion can be drawn from the sample.
Conditional Probability & Two-Way Tables
📌 Domain: Problem-Solving & Data Analysis · Probability and conditional probability · Two-variable data (two-way frequency tables)
Two-way table questions appear on nearly every Digital SAT. Identify the correct “given” row or column, then divide the joint count by that row/column total.
\(P(A)=\dfrac{\text{favourable}}{\text{total}}\). Conditional: \(P(A\mid B)=\dfrac{n(A\cap B)}{n(B)}\). Independence: \(P(A\mid B)=P(A)\) — knowing \(B\) gives no information about \(A\). Complement: \(P(\text{not }A)=1-P(A)\).
Q: Probability of preferring online, given Grade 11?Prefers online Prefers in-person Total Grade 11 48 72 120 Grade 12 32 48 80 Total 80 120 200
\(P(\text{online})=80/200=0.4\) and \(P(\text{online}\mid\text{Gr 11})=0.4\). These are equal, so grade and preference are independent in this dataset.
| Row label | Passed | Failed | Total |
|---|---|---|---|
| Studied | 54 | 6 | 60 |
| Did not study | 18 | 22 | 40 |
| Total | 72 | 28 | 100 |
Practice set
Fifty SAT-style problems across all four domains — algebra, advanced math, problem-solving & data, and geometry & trigonometry — tagged by difficulty. Click any problem for a full worked solution.
Solve \( 4x-3=13 \).
BasicIf \( y=2x-3 \), find \( y \) when \( x=4 \).
BasicA line has slope 3 and passes through \( (0,2) \). Find \( y \) when \( x=2 \).
Intermediate\( y=3x+2 \):
\[ 3(2)+2=8. \]If \( 2x+y=7 \) and \( x+y=4 \), find \( x \).
IntermediateSubtract the equations:
\[ x=3. \]Find the slope of the line through \( (1,2) \) and \( (5,10) \).
IntermediateSolve \( 2x+3=4x-7 \).
Intermediate\( 10=2x \):
\[ x=5. \]If \( 5x-2y=10 \) and \( x=4 \), find \( y \).
Intermediate\( 20-2y=10 \):
\[ y=5. \]A line is parallel to \( y=2x+1 \). What is its slope?
IntermediateParallel lines share a slope:
\[ 2. \]Solve \( 0.5x+1.5=4 \).
Advanced\( 0.5x=2.5 \):
\[ x=5. \]Find the x-intercept of \( y=2x-6 \).
IntermediateSet \( y=0 \): \( 2x=6 \):
\[ x=3. \]If \( f(x)=3x+1 \), find \( f(4) \).
IntermediateSolve the system \( 3x+y=7 \) and \( x+y=3 \).
AdvancedSubtract: \( 2x=4 \Rightarrow x=2 \), then \( y=1 \):
\[ x=2,\ y=1. \]The cost is \( C=5n+20 \). Find \( C \) when \( n=6 \).
IntermediateSolve \( x^{2}=49 \).
BasicFactor \( x^{2}+7x+12 \).
IntermediateSolve \( x^{2}+x-12=0 \).
Intermediate\( (x+4)(x-3)=0 \):
\[ x=3,\ -4. \]Find the vertex of \( y=(x+1)^{2}-4 \).
IntermediateHow many real solutions does \( x^{2}+2x+5=0 \) have?
AdvancedDiscriminant \( =4-20=-16<0 \):
\[ \text{none.} \]Expand \( (x+2)^{2} \).
IntermediateSolve \( 2x^{2}=18 \).
Intermediate\( x^{2}=9 \):
\[ x=\pm 3. \]Solve \( x^{2}+6x+9=0 \).
Advanced\( (x+3)^{2}=0 \):
\[ x=-3. \]Find the positive solution of \( x^{2}-9=0 \).
IntermediateFind the sum of the roots of \( x^{2}-7x+10=0 \).
AdvancedRoots \( 2 \) and \( 5 \) (or \( -\tfrac{b}{a} \)):
\[ 7. \]Find the x-intercepts of \( y=x^{2}-4 \).
IntermediateSolve \( x^{2}-8x+15=0 \).
Intermediate\( (x-3)(x-5)=0 \):
\[ x=3,\ 5. \]Find \( x^{2} \) when \( x=-5 \).
IntermediateSolve \( 3^{x}=27 \).
IntermediateSimplify \( x^{2}\cdot x^{3} \).
IntermediateSimplify \( (x^{3})^{2} \).
IntermediateIf \( 2^{x}=32 \), find \( x \).
AdvancedSimplify \( \dfrac{8x^{3}}{2x} \).
IntermediateA population of \( 100 \) doubles three times. Find the result.
Advanced\( 100\times2^{3} \):
\[ 800. \]Find \( 25\% \) of \( 80 \).
BasicShare \( 40 \) in the ratio \( 3:5 \). Find the larger part.
IntermediateEight parts, each \( 5 \); larger is 5 parts:
\[ 25. \]Find the mean of \( 5,\,10,\,15,\,20 \).
IntermediateFind the median of \( 4,\,8,\,15,\,16,\,23 \).
IntermediateMiddle of the ordered list:
\[ 15. \]What percent of \( 120 \) is \( 30 \)?
IntermediateA car travels at \( 60 \) mph for \( 2.5 \) hours. Find the distance.
IntermediateIncrease \( 80 \) by \( 25\% \).
IntermediateIf \( 4 \) pens cost \$6, find the cost of \( 10 \) pens.
AdvancedEach pen costs \$1.50:
\[ 10\times1.5=\$15. \]A fair die is rolled. Find \( P(\text{even}) \).
IntermediateFind the range of \( 3,\,7,\,2,\,9,\,5 \).
IntermediateA jacket costs \$50 after a \( 20\% \) discount. Find the original price.
Advanced\$50 is \( 80\% \) of the original:
\[ \frac{50}{0.8}=\$62.50. \]A car goes \( 150 \) miles on \( 3 \) gallons. Find the miles per gallon.
IntermediateFind the area of a rectangle \( 8\times5 \).
BasicA triangle has angles \( 30^{\circ} \) and \( 90^{\circ} \). Find the third.
IntermediateA right triangle has legs \( 5 \) and \( 12 \). Find the hypotenuse.
IntermediateFind the circumference of a circle with \( r=7 \). Take \( \pi=\tfrac{22}{7} \).
IntermediateEvaluate \( \sin 30^{\circ} \).
AdvancedFind the area of a triangle with base \( 10 \) and height \( 6 \).
IntermediateTest generator
Build a randomised test from this subject’s question bank. Pick the number of questions and a difficulty, generate, then reveal the worked answers when you’re ready — or print a clean copy to hand out.
Full official exam paper
Based on the 2024–2025 College Board Digital SAT: two adaptive math modules, 4 answer choices, plus student-produced responses (SPR). Calculator permitted throughout.
Downloads
Take the Digital SAT material offline. The PDF prints straight from your browser; a native Word packet with editable equation-editor math is available on request.
SAT notes & practice
All eight chapters plus every worked solution, formatted as a clean print document.
Native Word packet
Editable .docx with real Word equations (OMML) — ready for a timed drill or a printed practice module.
Past Papers & Official Tests
Officially released exams and mark schemes from the examining body.
Worksheet Library
10 original ClipSAT worksheets with full answer keys, organized by unit — free to download and print.