Front matter and how to use this book

Think Like a Top Scorer

D I G I T A L S A T M A T H P R E P

FIRST EDITION

Think Like a Top Scorer

Digital SAT Math Prep

Built for the top of the scale

by educators and top-scoring students

CollegePrepRoom Publication Series

CollegePrepRoom Publication Series

Take the Full Adaptive Practice Exam

The most accurate way to know where you stand is to take a Digital SAT Math practice exam that behaves like the real one. A full-length, adaptive-mode practice exam is available at collegepreproom.com, built to match official Digital SAT timing, structure, and module-to-module routing, so your practice conditions match test day.

Scan the code below, or visit the site directly, to take it.

collegepreproom.com

Best used after finishing this book, as a final check on where you stand before test day.

Think Like a Top Scorer: Digital SAT Math Prep

First Edition

Copyright © 2026 CollegePrepRoom Publication Series. All rights reserved.

Published by CollegePrepRoom Publication Series.

This book is an independent publication. SAT® is a trademark registered by the College Board, which is not affiliated with, and does not endorse, sponsor, review, or approve this book. All questions, explanations, figures, and instructional material in this book are original works created for it, and are not reproduced from any official SAT materials.

This book is intended for lawful, individual educational use. It is designed to help you build mathematical understanding, reasoning, and steady test-taking habits. No book can guarantee a particular score, score increase, admission, scholarship, or other result, and none is promised here; outcomes depend on the effort and circumstances of each reader.

The Digital SAT (its format, scoring, and policies) is set by the College Board and may change over time. While care has been taken to describe the test accurately as of this edition, the publisher makes no warranty that every detail remains current, and readers should confirm specifics through official College Board sources.

No part of this book may be reproduced, distributed, or transmitted in any form or by any means (electronic, mechanical, photocopying, recording, or otherwise) without the prior written permission of the publisher, except for brief quotations used in a review.

The publisher has taken care to make this book accurate and useful, but assumes no liability for any error, omission, or for any harm or damage claimed to arise from its use.

Contents

GETTING STARTED

How Digital SAT Math is organized 1

Two adaptive modules · how scoring works · question types · how to use this book

DIAGNOSTIC

Diagnostic Test 6

A 22-question diagnostic module · skills map by domain · worked solutions for every question

Answers, skills map, and solutions 17

PART I · ALGEBRA

Linear equations in one and two variables 24

One-variable equations · solution counts · literal equations · slope, intercepts, and points

Linear functions: rate of change and initial value 34

Rate and initial value · units · interpolation and extrapolation · building models from context

Systems of two linear equations 42

Elimination and substitution · linear combinations · solution counts · systems from context (strictly linear)

Linear inequalities in one and two variables 50

Solution sets · sign reversal · compound inequalities · absolute value · systems of inequalities

PART II · ADVANCED MATH

Equivalent expressions 60

Expanding and factoring · zeros from factored form · simplifying rational expressions

Nonlinear functions: notation, transformations, composition, and inverses 68

Function notation and graphs · composite and inverse functions · transformations · piecewise functions

Nonlinear functions: quadratic forms and features 78

Standard, factored, and vertex form · completing the square · axis of symmetry

Quadratic equations: real solutions and the discriminant 86

The discriminant · x-intercepts · when there are no real solutions · reading nonlinear graphs

Nonlinear equations in one variable 92

Factoring · square roots · the quadratic formula · radical equations · nonlinear systems and tangency

Nonlinear functions: exponential models 100

Rational exponents and radicals · growth and decay · percent change as a factor

PART III · PROBLEM-SOLVING & DATA ANALYSIS

Ratios, rates, proportional relationships, and units 110

Ratios and rates · unit conversion · density · part-to-part and part-to-whole

Percentages 117

Percent change · successive changes · percentage-point versus percent change

One-variable data: distributions, center, and spread 122

Mean, median, mode · standard deviation · IQR and box plots · outliers

Two-variable data: models and scatterplots 131

Scatterplots · line of best fit · residuals · association versus causation

Probability and conditional probability 139

Sample space · two-way tables · conditional probability · and/or/complement rules

Inference, margin of error, and statistical claims 146

Sampling · margin of error · random assignment versus random sampling · bias

PART IV · GEOMETRY & TRIGONOMETRY

Lines, angles, and triangles 154

Angle relationships · triangle angle-sum · the triangle inequality

Triangle similarity and congruence 160

Similarity criteria · proportional segments · congruence criteria · the SSA case

Right triangles and trigonometry 168

The Pythagorean theorem · special triangles · sine, cosine, and tangent

Circles 174

Circumference, arcs, and sectors · central and inscribed angles · tangent lines · circle equations

Area and volume 182

Area formulas · composite regions · surface area and volume · scale factors

Coordinate geometry across lines, distance, and circles 189

Distance and midpoint · slope for parallel and perpendicular lines · circle equations

Special right triangles and the unit circle 195

45-45-90 and 30-60-90 ratios · radians · unit-circle coordinates

PART V · EXECUTION

Desmos for equations, systems, functions, and data 205

Graphing equations and systems · reading zeros and extrema · sliders · regression

The two adaptive Math modules 218

How the adaptive format works · protecting Module 1 accuracy · marking for review

Back-solving and strategic test values 221

Testing answer choices · plugging in values · knowing when it proves versus suggests

Student-produced responses in Bluebook 225

Allowed formats · fractions and decimals · avoiding unsupported symbols

Pacing two 35-minute Math modules 228

Two-pass pacing · the five-minute checkpoint · correct-but-slow · checking the exact quantity asked

PART VI · DOMAIN MASTERY AND ADVANCED PRACTICE

Four domain mastery sets and one advanced Module 2 practice set, followed by the full-length benchmark.

Domain mastery set A: Algebra 235

Domain mastery set B: Advanced Math 237

Domain mastery set C: Problem-Solving and Data Analysis 239

Domain mastery set D: Geometry and Trigonometry 242

Advanced Module 2 practice 246

PART VII · BENCHMARK TEST

One full-length fixed-form benchmark with two 22-question modules, a complete answer key, skills map, and worked solutions.

Benchmark Test (Modules 1 & 2) 252

Answers, skills map, and solutions 276

REFERENCE & TEST DAY

The SAT Math reference sheet: provided formulas and required knowledge 285

What Bluebook gives you · the formulas and facts you still need to know cold

Digital SAT test day: device, Bluebook, break, and Math tools 287

Device and Bluebook setup · the break · the calculator and reference tools · if something goes wrong

Use your diagnostic and benchmark to identify remaining Math gaps 290

Comparing diagnostic and benchmark results · diagnosing content vs. execution errors · a final review plan

Final Math review by domain 292

A one-page summary of every domain, ready for the night before

Preface

SAT Math tests a defined set of mathematics, but knowing the topic list is only the beginning. Students must recognize relationships, choose useful representations, interpret quantities, build equations from information, and decide which method fits the problem.

In Algebra, linear equations can be presented as equations, graphs, tables, or contexts. Students may need to find or interpret slope and intercept, construct a linear function from given information, or determine how one quantity changes with another. Systems can involve an intersection, the number of solutions, or a constant that produces one solution, no solution, or infinitely many solutions. Inequalities can require translating a constraint into mathematics and interpreting the resulting solution set.

Part I develops these forms of Algebra.

Advanced Math places greater emphasis on structure. Equivalent expressions must sometimes be rewritten into a form that reveals the needed information. A quadratic can appear in standard, factored, or vertex form. Students may need its zeros, vertex, maximum or minimum, coefficients, discriminant, or number of real solutions. An unknown constant may be included, with its value determined from a condition imposed on the equation.

Functions may appear as equations, graphs, or tables. Tested skills include function notation, transformations, composition, inverse relationships, and interpretation of parameters. Exponential relationships include growth and decay, percent change, growth factors, equivalent forms, and different time intervals. Polynomial, rational, radical, and absolute-value relationships are also part of the tested content.

Part II covers these Advanced Math skills and the different forms in which they can appear.

Problem-Solving and Data Analysis requires precise interpretation. Ratios, rates, proportional relationships, unit conversions, percentages, and scale must be translated correctly before calculation. Successive percentage changes require multiplicative reasoning. The original or reference quantity matters.

Data questions can involve mean, median, range, interquartile range, standard deviation, distributions, scatterplots, lines and curves of best fit, and residuals. Probability includes conditional probability and two-way tables. Statistical reasoning extends to sampling, margin of error, and the conclusions supported by a study. Random sampling affects whether results can be generalized to a population. Random assignment affects whether a causal conclusion can be supported.

These skills are developed in Part III.

Geometry and Trigonometry includes lines and angles, triangles, similarity and congruence, the Pythagorean theorem, special right triangles, sine, cosine, tangent, radians, the unit circle, circles, coordinate geometry, area, surface area, and volume.

The required relationship is not always supplied directly. A missing length may have to be established before area can be calculated. Similarity can connect several lengths. A right triangle may require selecting the appropriate trigonometric relationship. Circle problems can connect radii, tangent lines, angles, arcs, sectors, and coordinate equations. Changes in dimensions can affect length, area, and volume differently.

Part IV develops these relationships and combines them in progressively more demanding problems.

This combination of skills is especially important on the Digital SAT. Higher-difficulty questions can retain familiar mathematics while changing how much of the structure is immediately visible. A numerical value may become a parameter. Information about the solutions of an equation may be used to determine a coefficient. A graph may carry information that would ordinarily be given algebraically. Geometry may produce an algebraic relationship. A contextual problem may first require a mathematical model. Several familiar steps may need to be connected before the requested quantity can be found.

Recent test-taker reports are consistent on an important point even though individual SAT forms differ. Students frequently describe the later portion of a higher-difficulty second Math module as the point where familiar concepts become less direct, several relationships may need to be connected, and time becomes more consequential. The best preparation is therefore broad enough to cover the complete content and deep enough to handle changes in presentation.

The structure of this book follows that goal. Chapter practice establishes each skill. Later problems vary the representation through equations, graphs, tables, diagrams, contexts, and parameters. Domain Mastery sets mix material without a chapter heading identifying the method. Advanced Module 2 practice concentrates higher-level transfer. The final benchmark brings the domains together under SAT timing.

Desmos is integrated throughout the preparation. Graphs can expose zeros and intersections. Tables can reveal function values. Regression can model data. Algebra remains faster for many problems. Part V develops the judgment to choose an efficient method and to combine calculator and hand work when useful.

The purpose of this book is not to predict individual SAT questions. It is to cover the tested mathematics completely and practice it in enough forms that a change in presentation does not create a new mathematical problem.