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

Use the skills map to return to a worked example, then try a fresh question and record what you will change next time.

Strategy quick reference

1. Name the target; then choose the representation

Target: a variable, an expression, a count, a coordinate, a unit-bearing quantity, or a justified conclusion.

Conditions: nonzero denominators, nonnegative radicals, positive dimensions, whole-number counts, intervals, and the actual comparison group.

Clue in the problem	First method to consider	Validity check
A repeated group resembles the target	Replace the group directly, or form a linear combination.	Did you return the target rather than an intermediate variable?
A percentage process has multiple stages	Use a multiplier for each stage.	What is the base at each stage?
A mean changes when data are added	Reconstruct the old and new totals.	Did the count change too?
Equal travel distances at different speeds	Add distances; separately add times.	Average speed is total distance divided by total time.
Several numerical answer choices	Test candidates in the original model.	Is the candidate the quantity asked for?
Equivalent-expression choices	Choose allowed, informative values; eliminate mismatches.	One agreement is not a universal proof.
A factor, vertex, or root sum is visible	Preserve the form that reveals the needed feature.	Is expansion helping or hiding structure?
A parameter controls solution count	Classify degree, discriminant, domain, and signs.	Did you check boundary and reduced-degree cases?
An approximate nonlinear intersection	Graph a simplified, correct system.	Check domain, window, and requested coordinate.
A whole-number threshold	Locate the boundary, then test neighboring integers.	Did every original constraint survive?

The two-sentence finish

“This value satisfies the original conditions because....” Then: “The question asks for..., so the response is....” A second calculation is most useful when it tests a different risk, not when it repeats the same assumptions.

2. Keep structure and conditions together

This is a study reference, not a reproduction of the test’s supplied formula sheet.

Relationship	Strategic use and condition
$A 2 - B 2 = (A - B)(A + B)$	Avoid computing two large squares.
$A 2 + B 2 = (A + B) 2 - 2AB$	Find a symmetric expression from a sum and a product.
$\frac{1}{r}$	Use only when $rs \neq 0.$
$ax 2 + bx + c = 0$	For $a \neq 0,$ root sum is $-b/a,$ product is $c/a,$ and discriminant is $b 2 - 4ac.$
$y = a(x - h) 2 + k$	Vertex is (h, k). Equal outputs at distinct inputs have input sum $2h,$ when $a \neq 0.$
$Q new = Q old (1 \pm p/100)$	Use the sign appropriate to the change; compose successive multipliers.
$total = mean \times count$	Pool totals and counts before computing a combined mean.
$N(t) = ab t$	For $a, b > 0,$ $N(t 2)/N(t 1) = b t 2 -t 1 .$ Ratios remove the initial factor.
Length factor $s > 0$	Similar-figure areas scale by $s 2;$ similar-solid volumes scale by $s 3 .$
$\frac{P(A ∩ B)}{P(B)}$	Requires $P(B) > 0.$ In a count table, restrict the denominator to group B.

The parameter checklist

Degree first. Can a leading coefficient become zero? Analyze that value in the original equation before using a quadratic rule.

Count second. For a genuine quadratic, $discriminant > 0,$ $= 0,$ or $< 0$ means two, one, or zero distinct real roots. For systems, these roots must still produce valid intersection points.

Restrictions third. Two positive real roots require real roots and positive sum and product. Squaring and clearing denominators may introduce invalid candidates. An input endpoint counts only when it belongs to the stated domain.

Boundaries last. Test equality cases explicitly. When the parameter itself must be an integer, first find the valid real intervals, then select or count the allowed integers.

Skills-to-examples map

Use the clue column to diagnose a missed problem. A topic name such as “quadratic” is less actionable than “I used the discriminant without checking the leading coefficient.” Each example reference is clickable.

Section 1: Build a mathematical model

Clue or skill	Worked examples
Fees, tax, and successive changes	1.01, 1.04
Ratios, totals, and mean reconstruction	1.02, 1.03, 1.13
Rates, average speed, and simultaneous work	1.05, 1.06, 1.11
Conservation and changing quantities	1.07, 1.09, 1.12
Two-stage groups and conditional selection	1.10, 1.14
Geometry models and a maximum under constraints	1.08, 1.15

Section 2: Test intelligently

Clue or skill	Worked examples
Percent or scale-factor expressions	2.01, 2.02, 2.03
Numerical candidates and target qualifiers	2.04, 2.05, 2.11
Restrictions and uninformative inputs	2.06, 2.07, 2.08
Models, difference quotients, and comparison bases	2.09, 2.10, 2.13
Universal claims, extraneous roots, and identities	2.12, 2.14, 2.15

Section 3: Preserve useful structure

Clue or skill	Worked examples
A known group or a useful linear combination	3.01, 3.02, 3.03
Factoring, substitution, and a derived target	3.04, 3.05, 3.06
Root relationships and symmetric expressions	3.07, 3.09, 3.10, 3.14
Vertices, common factors, and equal outputs	3.08, 3.11, 3.12
Quadratic-in-a-group and a supplied factor	3.13, 3.15

Section 4: Classify parameters and solutions

Clue or skill	Worked examples
Linear identities, contradictions, parallel lines	4.01, 4.02, 4.03
Coefficient comparison and a supplied root	4.04, 4.05
Discriminants and tangency	4.06, 4.07, 4.08, 4.09
An exponential model from two observations	4.10
Reduced-degree cases and excluded inputs	4.11, 4.12
Radical boundaries, signs, and integer parameters	4.13, 4.14, 4.15

Section 5: Audit the result

Clue or skill	Worked examples
Compound units and squared conversions	5.01, 5.02
Percent bases and conditional denominators	5.03, 5.04, 5.08
Physical roots, squaring, and cancellation	5.05, 5.06, 5.07
Trigonometric role and angle-mode checks	5.09, 5.13
Integer thresholds and all-constraint checks	5.10, 5.15
Estimates, weighted groups, and bounds	5.11, 5.12, 5.14

Section 6: Manage method and time

Clue or skill	Worked examples
Short direct calculations and elimination	6.01, 6.02, 6.03
Approximate intersections and finite candidates	6.04, 6.05
Algebra rather than unnecessary tool setup	6.06, 6.07, 6.09
Exact parameter cases and a final numerical step	6.08, 6.10
Workflow: completion budgets and deferring	6.11, 6.12, 6.13
Workflow: changing a response and reviewing practice	6.14, 6.15

A useful return path

Missed a mixed question? Read its explanation, attempt the linked example again with the solution covered, and then try a fresh problem with the same decision but different surface details. Recalling an answer is not the same as recognizing a method.

Review laboratory

Turn an error into a specific next action

Use one primary code for the earliest error. Later arithmetic often fails because the initial representation was wrong; repairing only the final calculation will not fix that pattern.

Code	Diagnosis	Useful replacement habit
T	Wrong target	Write the exact requested expression or unit before solving.
M	Wrong model	Label quantities and translate one relationship at a time.
S	Inefficient structure	Look for a known group, a ratio, or a useful equation combination.
C	Missing condition	List excluded inputs and parameter exceptions before transforming.
E	Execution error	Identify the specific sign, distribution, arithmetic, or tool-entry failure.
V	Weak verification	Check an original equation, a unit, a bound, or an adjacent integer.
P	Pacing or navigation	Leave a restart line; separate a flag from an entered response.

One deliberate practice cycle

Attempt. Select a fresh set and work independently. Keep original scratch work. If timed, record actual elapsed time rather than an impression of being fast or slow.

Diagnose. Review wrong answers, guesses, and slow correct answers. Record the first decision you would change. “Be more careful” is not specific enough to practice.

Rebuild. Solve a missed problem from a blank page. Explain why the method applies, including its restrictions. Then compare an alternative when one offers a genuinely different route.

Transfer. Use a fresh problem that changes the story, numbers, representation, or target. The official Student Question Bank allows targeted selection by domain, skill, and difficulty. ^[12]

Rehearse. After repairing the conceptual issue, test the revised behavior under the timing and tools you expect to use. Full-length Bluebook practice supports digital rehearsal; this guide’s fixed question sets are not equivalent to a full adaptive test. ^{[7] [10]}

Write an observable next action

Instead of “I will get faster at quadratics,” write “Before using the quadratic formula, I will check whether the target follows from the root sum, the product, or the vertex.” Evaluate that behavior on fresh questions, not only on ones you remember.

Printable strategy error log

Date: __________

Practice source / set: __________

Record method quality separately from correctness. Reuse this page for missed, guessed, and slow-correct questions.

REVIEW ENTRY 1

Question ID: __________

Correct / wrong / guessed / slow-correct

Primary code (T, M, S, C, E, V, P): __________

Time, if recorded: __________

The actual target and restrictions:
__________

The first decision or step to change:
__________

A better method and an independent check:
__________

Next fresh practice / date: __________

REVIEW ENTRY 2

Question ID: __________

Correct / wrong / guessed / slow-correct

Primary code (T, M, S, C, E, V, P): __________

Time, if recorded: __________

The actual target and restrictions:
__________

The first decision or step to change:
__________

A better method and an independent check:
__________

Next fresh practice / date: __________

One behavior to test next session: __________

Official sources and series connections

Official sources and edition notes: 1–6

Verified September 2026. The linked titles below lead to primary College Board sources. Numbered references in the lessons point to this list. Testing policies and software details can change; confirm the current official guidance and any approved accommodations before test day.

Scope of attribution. Sources support testing facts, content scope, and official practice descriptions. The original examples, mathematical explanations, diagrams, difficulty labels, timing scenarios, and study recommendations are this guide’s instructional content. No official question has been reproduced, and no score-improvement guarantee is made.

[1] How the SAT Is Structured. Standard section timing, module organization, and the relationship between first- and second-module difficulty. Used in the pacing discussion.

[2] The Math Section: Overview. The four official Math domains and the role of contextual applications. The six strategy sections in this guide are cross-domain teaching categories, not additional official domains.

[3] Bluebook Testing Tools. Timer, calculator, question menu, Mark for Review, and option-elimination descriptions. The current student guide in source 11 gives SAT-specific tool details.

[4] SAT Suite of Assessments Calculator Policy. Permitted calculator use and restrictions. Calculator availability does not imply that every problem benefits from calculator use. Review the current policy for any personal calculator.

[5] Student-Produced Responses. The numerical-response format and the distinction between a question with multiple acceptable responses and the one response a student supplies.

[6] How Are Scores Calculated? Adaptive design, review within a module, scoring cautions, and College Board’s qualified guidance favoring an informed guess over a blank for most students trying their best.

Keep the claims separate

A timing suggestion is an adjustable study choice. A calculator restriction is a testing rule. A mathematical identity requires valid assumptions and reasoning. None should be treated as evidence for the others.

Official sources: 7–12

[7] Practice on Bluebook. Distinguishes untimed test previews without answer feedback from full-length practice. Describes scored practice when signed in and accommodation selection for rehearsal.

[8] Advanced Math. Official scope for equivalent expressions, nonlinear equations and systems, and nonlinear functions. Helps set the boundary of the challenging algebraic applications.

[9] Problem-Solving and Data Analysis. Official scope for rates, percentages, data, probability, inference, and evaluation of statistical claims. Supports the inclusion of denominator, weighting, and interpretation checks.

[10] Full-Length SAT Suite Practice Tests. Official full-length practice resources and the distinction between digital adaptive practice and nonadaptive paper materials.

[11] SAT test-day information and student guide. College Board’s test-day resource page provides current preparation information and access to its student guide. For review tools and the timer alert, see source 3; for scoring, see source 6.

[12] How to Use the Student Question Bank. Describes selecting official practice by assessment, test, domain, skill, and difficulty. Useful for finding fresh questions after reviewing an error pattern.

How this volume connects to the series

Guides 1–5 establish foundations and the four content domains. Return to those volumes when an unfamiliar rule, not a strategic decision, is blocking progress.

Guide 6, Calculator and Digital Testing Skills, develops tool execution and digital workflow. This volume emphasizes when to choose a tool, when to avoid unnecessary setup, and how to verify the result.

Guide 7 is the synthesis volume: name the target, build or recognize a relationship, choose a justified method, and check that the answer survives the original conditions.

The habit to carry forward

Do not ask only, “Can I solve this?” Ask, “What makes this method valid here, what can go wrong, and what is the shortest check that tests the actual risk?”