Review and reference

Worksheets and ledgers are optional reference records. Write on paper or keep your own notes; these records do not save progress on this website.

On this page

The practice question numbers below are the stable Guide question numbers shown on each question, even when the bank shuffles their order.

Arithmetic and algebra reference

Quick reference / arithmetic and algebra

This is a learning reference, not a copy of the test’s formula sheet. Know the meaning and conditions of each rule; a remembered symbol pattern without those conditions is unreliable.

Idea	Relationship and condition
Order of operations	Grouping; powers; multiplication and division left to right; addition and subtraction left to right. A fraction bar groups both numerator and denominator.
Signs and powers	$-a 2 = -(a 2);$ $(-a) 2 = a 2 .$ Parentheses determine the base. $a - (-b) = a + b.$
Fraction addition	$\frac{a}{b}$ for b, $d \neq 0.$ A common denominator makes the parts the same size.
Fraction multiplication	$\frac{a}{b}$ for b, $d \neq 0.$ Cancel common nonzero factors, not terms of a sum.
Fraction division	$\frac{a}{b}$ for b, c, $d \neq 0.$ Invert the divisor only.
Exponent rules	$a m a n = a m+n;$ $a m /a n = a m-n$ for $a \neq 0;$ $(a m) n = a mn .$ These forms are safe for integer exponents where defined. For fractional exponents, check the real domain.
Zero/negative powers	$a 0 = 1$ and $a -n = 1/a n$ for $a \neq 0.$ A negative exponent means reciprocal, not a negative value.
Square roots	$\sqrt{a}$ for $a \geq 0;$ $\sqrt{x^{2}}$ The equation $x 2 = a > 0$ has solutions $\sqrt{a}$
Scientific notation	$a \times 10 n,$ where $1 \leq \|a\| < 10$ and n is an integer, for nonzero numbers. Multiply coefficients and combine powers, then normalize.
Distribution	$a(b + c) = ab + ac.$ The signed multiplier applies to every term in the group.
Difference of squares	$a 2 - b 2 = (a - b)(a + b).$ Use it to factor; keep any original denominator restrictions.
Equality	Apply the same operation to both sides. Division requires a nonzero divisor. Squaring can introduce extra solutions, so check the original.

Before a calculation becomes long

Can you simplify exact fractions first? Can you substitute a given group directly? Can you check the sign and rough size? Use a calculator when it reduces work without hiding the structure you need to understand.

Functions and modeling reference

Idea	Relationship or decision
Function notation	f (a) supplies the input a; evaluate the rule. $f(x) = b$ supplies the output b; find the input or inputs.
Function definition	Each allowed input has one output. Several inputs may share an output.
Intercepts	Vertical: set $x = 0.$ Horizontal: set $y = 0.$ A zero is an input whose output is zero.
Domain	Exclude zero denominators. Require nonnegative radicands for even real roots. Apply context restrictions as well.
Linear rate	$m = (y 2 - y 1)/(x 2 - x 1)$ for different inputs. Rate units are output units per input unit.
Linear/proportional	$y = mx + b$ is linear; $y = kx$ is proportional. A finite set of collinear points does not prove an unspecified global rule is linear.
Ratio to total	If $A : B = p : q,$ then $A/(A + B) = p/(p + q)$ when the positive quantities form the whole.
Percent part	$Part = (p/100)\times base.$ $Percent number = 100(part/base)$ for nonzero base.
Percent change	$New = original (1 \pm p/100).$ Recover the original by dividing by the applicable nonzero multiplier.
Rate model	$Amount = rate \times time$ when the rate is constant and units match. Fixed fee plus repeated cost: $C = b + rn.$
Mean	$Mean = total/count$ for a positive count. Therefore $total = count \times mean.$
Constraints	At least: $\geq;$ at most: $\leq.$ Multiplying or dividing an inequality by a negative number reverses its direction.
Whole items	Under a maximum budget, choose the greatest permitted integer. To meet a minimum need, choose the least sufficient integer. Test an adjacent integer.
Units and precision	Cancel units explicitly. Square conversion factors for area; cube them for volume. Keep exact values until the final requested rounding step.

The five-part answer check

Target: Did I report the requested quantity? Sign: Is it permitted? Size: Is it plausible? Units: Do they match? Restrictions: Does the value fit the original domain and context?

Skills map: orientation and arithmetic

Each range below names a focused review cluster. Click an endpoint to open that example. Use a cluster only when the diagnostic or independent work shows that it is needed.

Worked examples	Skills to retrieve
Example 1.01–Example 1.03	Target versus intermediate value; checking candidates; exact numeric entry.
Example 1.04–Example 1.06	Repeating-decimal precision; one permitted value; calculator grouping.
Example 1.07–Example 1.09	Method choice and graph checks; scaled units; estimation and magnitude.
Example 1.10–Example 1.12	First invalid step; principal roots versus equation solutions; median versus mean.
Example 1.13–Example 1.15	A multistep target; a return-and-review plan; secure accuracy versus guesses.
Example 2.01–Example 2.03	Signed arithmetic; grouping and powers; greatest common factor and least common multiple.
Example 2.04–Example 2.06	Common denominators; mixed-number subtraction; fraction products and cancellation.
Example 2.07–Example 2.09	Fraction division; compound fractions; decimal division.
Example 2.10–Example 2.12	Fraction-decimal-percent equivalence; negative-number order; scientific notation.
Example 2.13–Example 2.15	Exact roots; avoiding early rounding; magnitude plus calculator verification.

A three-question oral check

Explain why $-3 2$ and $(-3) 2$ differ. Explain why $3/4 \div 1/2$ is greater than $3/4.$ Explain why a correct value of x can still be a wrong final response. If your explanation relies only on a memorized slogan, return to the meaning of the operation.

For students who already calculate fluently

Do not repeat an entire arithmetic chapter simply to accumulate completed lessons. Attempt its exit questions, explain the vulnerable steps, and move on when the evidence is secure. Keep a short list of any recurring sign, precision, or entry errors for targeted review.

Skills map: algebra and functions

Worked examples	Skills to retrieve
Example 3.01–Example 3.03	Terms and signed coefficients; negative substitution; grouping an entire numerator and denominator.
Example 3.04–Example 3.06	Like terms; signed distribution; factoring a common factor.
Example 3.07–Example 3.09	Dividing a grouped expression; canceled factors and restrictions; inverse operations.
Example 3.10–Example 3.12	Equations with distribution; one, none, or infinitely many solutions; formula rearrangement.
Example 3.13–Example 3.15	Finding a parameter; substituting an entire group; a counterexample to a false identity.
Example 4.01–Example 4.03	Coordinate order; axis scales; evaluating a function at a negative input.
Example 4.04–Example 4.06	Reading tables; finding an input for an output; zeros and intercepts.
Example 4.07–Example 4.09	Rates with unequal input intervals; detecting nonlinearity; the function definition.
Example 4.10–Example 4.12	Discrete contextual domains; algebraic restrictions; open and closed endpoints.
Example 4.13–Example 4.15	Equal function values; piecewise boundaries; recognizing insufficient information.

Contrast pairs worth explaining

An expression names a quantity; an equation asserts equality. An identity holds for all allowed inputs; a solution makes a particular equation true. A domain lists inputs; a range lists outputs. An algebraically allowed input may still be disallowed by the context. These distinctions become more important, not less, as problems become more advanced.

What counts as a convincing check?

Substitution can verify a proposed equation solution. A counterexample can disprove an identity. One successful numerical substitution cannot prove an identity for every real input. A small graph window cannot prove that no other intersection exists outside that window. Match the strength of your check to the claim you are making.

Skills map: modeling and next steps

Worked examples	Skills to retrieve
Example 5.01–Example 5.03	Reversed comparison language; grouping a sum; comparing two quantities within a total.
Example 5.04–Example 5.06	Part-part ratios; percent increase; reversing a percent decrease.
Example 5.07–Example 5.09	Fixed plus variable cost; distance with a time conversion; converting a compound rate.
Example 5.10–Example 5.12	Maximum affordable count; minimum required count; perimeter as a route to area.
Example 5.13–Example 5.15	Two-category revenue; a fraction of a remainder; a multistep purchase with whole packs.

Carry these foundations into the domain volumes

The following connections are a study sequence, not additional official domains. Move into a domain while continuing targeted foundation review when needed.

Next volume	Foundations that do the most work
Algebra	Signed distribution, reversible equation steps, rates and intercepts, fixed-versus-variable models, and whole-item inequalities.
Advanced Math	Factoring, exponent notation, exact roots, domain restrictions, function inputs and outputs, and the difference between a check and a proof.
Problem-Solving and Data Analysis	Fractions and percents, comparison bases, ratios, compound units, graph scales, and a clearly identified requested statistic.
Geometry and Trigonometry	Exact fractions and radicals, formula rearrangement, coordinate order, dimension checks, and length-versus-area-versus-volume scaling.

A final translation test

Before solving a word problem, read your equation aloud. Point to the phrase that justifies each term. After solving, test the result against the original sentence. An equation that is easy to solve but does not match the story is not a useful model.

Readiness and next steps

Route A / Audit and advance. When the diagnostic is secure across a cluster, skim its reference rules and attempt the section exit. If all five exit answers are independently correct and explainable, begin the relevant domain work. Keep any slow or uncertain item in the error log.

Route B / Repair a focused gap. When one cluster is weak, study only the relevant concepts and examples. Attempt the exit with the lesson closed, then use a mixed set to see whether you recognize the method without a topic label. Revisit the skill in a later session rather than immediately repeating a memorized answer.

Route C / Rebuild the chain. When several clusters are weak, proceed through Sections 2–5 in order while using Section 1’s scratch-work routine. Start with reliable untimed work. Add time pressure after you can explain the method, and continue into the domain volumes rather than treating foundations as an indefinite prerequisite course.

Suggested section-exit decisions, not score cutoffs

5 of 5, all secure: move forward and check transfer in mixed practice.

4 of 5, or any correct guess: repair the specific gap and solve a fresh parallel item.

3 or fewer secure: revisit the concept cluster and explain the first failed step before adding speed. Five questions are a small sample. These are practical decisions for this book, not validated mastery thresholds or SAT score predictions.

What readiness looks like in your work

You can define the unknown and target before calculating. You can preserve signs and grouping through a multistep expression. You can interpret a rule, table, graph, or sentence without reversing input and output. You can explain a unit conversion and a percent base. You can reject a result that violates a domain or whole-item restriction. You can verify a method on a fresh item, not only on the example you just read.

From a booklet to test conditions

Use Bluebook’s preview to practice the interface; use a scored full-length adaptive practice test to examine whole-test performance. Review results by skill rather than studying only the total score. The preview and full-length tests serve different purposes.^[7]

Before an actual administration, confirm your testing arrangements and current device requirements, complete exam setup, and follow your admission-ticket instructions. Bring a charged approved testing device and check the current calculator policy rather than relying on an old model list. Tell the proctor promptly about a technical problem.^[5][9]

Make an error useful

Date: Question/source:

My answer: Correct answer:

Confidence before checking: secure / uncertain / guessed

Primary code: K / R / S / E / T / P Secondary code:

Question to answer	Your record
What quantity was actually requested?
What was my first invalid or missing step?
Which rule or interpretation repairs it?
Reconstruct a correct method without copying.
What specific habit will prevent the error?
Fresh retest item, later date, and result

$K = knowledge;$ $R = reading;$ $S = structure;$ $E = execution;$ $T = tool/entry;$ $P = pacing.$ A useful habit is observable: “parenthesize a negative substitution,” not “try harder.” Copy this worksheet or keep your own notes for additional review records.

Sources, scope, and editorial notes

Test-format and policy statements were checked against the official College Board resources below for this September 2026 edition. Current directions, your approved accommodations, and your administration’s instructions take precedence over any study guide. Click a linked source title to open it.

Official sources / structure and response conventions

[1] College Board. How the SAT Is Structured. Standard section timing, question counts, module adaptivity, and the break between sections.

[2] College Board. Math Specifications. The four official Math domains, approximate content distribution, and operational/pretest questions per module.

[3] College Board. Student-Produced Responses. Multiple-choice versus numeric-response formats and entering one valid answer when several are possible.

[4] College Board. SAT Practice Test 11, Math reference and directions. Numeric-entry character limits, improper-fraction and decimal conventions, and test reference information. This nonadaptive test is cited here for its directions, not as the source of digital module question counts.

Original work and scope

All 75 worked examples and all 79 independent questions in this guide were written for this volume. Their contexts, numerical values, graphs, tables, explanations, and instructional difficulty labels are not official College Board items or calibrated ratings. No official practice questions have been reproduced.

The independent questions comprise a 24-question diagnostic, 25 section exits, and 30 mixed-practice items. They intentionally include some explanation and translation tasks that are useful for learning but do not imitate the SAT’s exact response formats. Section 1’s last two worked examples are explicitly labeled planning scenarios.

This foundations volume supports, but does not replace, the four domain guides. Higher-level quadratics, nonlinear systems, inference, advanced geometry, and full calculator strategy are developed in those volumes. Elementary rules included here are not a claim that every prerequisite appears as a standalone SAT question.

Tools, practice, and scoring

[5] College Board. SAT Suite of Assessments Calculator Policy. Embedded calculator options, current handheld-calculator restrictions, and prohibited CAS functionality. Recheck before the administration you will take.

[6] College Board. Bluebook Testing Tools. Timer, question navigation and review tools, option elimination, calculator, and Math reference access.

[7] College Board. Full-Length Digital Practice Tests on Bluebook. The distinction between an unscored test preview and full-length practice, with results and review in My Practice.

[8] College Board. How Are Scores Calculated?. Response patterns, item characteristics, unscored questions, and why a simple raw-count conversion is not a sound score model.

[9] College Board. What to Expect on Test Day. Module navigation, exam setup, device readiness, and seeking proctor help during testing.

Editorial boundaries

Exactness. Unless a prompt requests an approximation, examples preserve exact fractions and radicals when useful. Rounded displays are identified as approximations. Numeric-entry conventions are treated separately from ordinary calculation precision.

Evidence. Proposed study routes, pacing trials, confidence labels, and readiness decisions are instructional suggestions. They are not validated interventions, promises of score improvement, or substitutes for a properly administered assessment.

Verification. Mathematics is established by the explanation: valid transformations, explicit domains, unit checks, counterexamples, and substitution into original conditions. A calculator display or a graph’s appearance alone is not treated as a proof of a general claim.

The foundation to keep

A strong solution preserves meaning from the first sentence to the final answer. Know what is being asked, choose a model you can explain, and check that the result still belongs to the original problem.

Answers and explanations

Use the answer first as a checkpoint, not as a substitute for a method. A correct response that you cannot explain remains a review item. Each solution stays with its question. References point to a worked example that repairs the relevant skill.