SAT · ACT · STANDARDIZED TEST PREP
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SAT / ACT Prep —
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Structured test prep from a PhD engineer who understands how standardized tests are designed. No tricks — just mastery.
SAT Math
Heart of Algebra
The Heart of Algebra domain tests your ability to analyze, solve, and create linear equations and systems. Roughly 33% of SAT Math questions fall in this domain — the highest of any category.
Key Formulas & Concepts
- Linear equations: y = mx + b (slope-intercept form)
- Systems of equations: substitution and elimination
- Absolute value equations: |ax + b| = c gives two cases
- Linear inequalities and graphing solution regions
- Interpreting slope and y-intercept in context
Test Strategy
- For "no solution" systems: parallel lines have equal slopes
- For "infinitely many solutions": identical equations
- When solving |x| = k, set up x = k AND x = −k
- Check extraneous solutions in absolute value equations
- Pick numbers to verify algebraic answers when stuck
Common Mistakes
- Sign errors when distributing a negative across parentheses
- Forgetting to flip the inequality when dividing by a negative number
- Not checking both cases when solving absolute value equations
- Misidentifying slope from standard form (Ax + By = C → slope = −A/B)
Practice Problem 1
If 3x − 2(x + 4) = 5x + 8, what is the value of x?
Show Answer & Solution
Distribute: 3x − 2x − 8 = 5x + 8
Combine like terms: x − 8 = 5x + 8
Subtract x from both sides: −8 = 4x + 8
Subtract 8: −16 = 4x
x = −4
Combine like terms: x − 8 = 5x + 8
Subtract x from both sides: −8 = 4x + 8
Subtract 8: −16 = 4x
x = −4
Practice Problem 2
The system below has no solution. What is the value of k?
2x + ky = 6
4x + 8y = 5
2x + ky = 6
4x + 8y = 5
Show Answer & Solution
For no solution, the lines must be parallel: equal slopes, different y-intercepts.
Slope of line 1: −2/k | Slope of line 2: −4/8 = −1/2
Set equal: −2/k = −1/2 → k = 4
Verify: if k = 4, line 1 becomes 2x + 4y = 6 (i.e., x + 2y = 3) and line 2 is 4x + 8y = 5 (i.e., x + 2y = 5/4). Parallel, different intercepts → no solution. k = 4
Slope of line 1: −2/k | Slope of line 2: −4/8 = −1/2
Set equal: −2/k = −1/2 → k = 4
Verify: if k = 4, line 1 becomes 2x + 4y = 6 (i.e., x + 2y = 3) and line 2 is 4x + 8y = 5 (i.e., x + 2y = 5/4). Parallel, different intercepts → no solution. k = 4
Practice Problem 3
|2x − 3| = 11. What are all possible values of x?
Show Answer & Solution
Case 1: 2x − 3 = 11 → 2x = 14 → x = 7
Case 2: 2x − 3 = −11 → 2x = −8 → x = −4
x = 7 or x = −4
Case 2: 2x − 3 = −11 → 2x = −8 → x = −4
x = 7 or x = −4
SAT Math
Passport to Advanced Math
This domain covers the algebraic skills needed for higher-level mathematics. Questions involve quadratics, polynomials, rational expressions, radical equations, and function notation. About 28% of SAT Math.
Key Concepts
- Factoring quadratics: x² + bx + c = (x + p)(x + q)
- Quadratic formula: x = [−b ± √(b²−4ac)] / 2a
- Vertex form: f(x) = a(x−h)² + k
- Polynomial long division and remainder theorem
- Rational expressions: simplifying and solving
- Radical equations: isolate the radical, then square
- Function notation: f(g(x)) and transformations
Test Strategy
- When a quadratic has two solutions in context, check both for plausibility
- Completing the square reveals vertex and minimum/maximum directly
- Extraneous solutions arise after squaring — always verify
- For f(g(x)), evaluate g(x) first, then plug into f
- Use the discriminant (b²−4ac) to determine number of real roots without solving
Common Mistakes
- FOIL errors: (a + b)² ≠ a² + b² — the cross term 2ab is required
- Forgetting ± when taking a square root: √x² = |x|, not just x
- Canceling terms instead of factors in rational expressions
- Applying function transformations in the wrong order or direction
Practice Problem 1
What are the solutions to x² − 5x − 14 = 0?
Show Answer & Solution
Factor: find two numbers that multiply to −14 and add to −5. Those are −7 and +2.
(x − 7)(x + 2) = 0
x = 7 or x = −2
(x − 7)(x + 2) = 0
x = 7 or x = −2
Practice Problem 2
The function f is defined by f(x) = 2x + 3. What is f(f(4))?
Show Answer & Solution
First, f(4) = 2(4) + 3 = 11
Then, f(f(4)) = f(11) = 2(11) + 3 = 25
Then, f(f(4)) = f(11) = 2(11) + 3 = 25
Practice Problem 3
Solve for x: √(3x + 1) = x − 1
Show Answer & Solution
Square both sides: 3x + 1 = (x − 1)² = x² − 2x + 1
Rearrange: 0 = x² − 5x
Factor: 0 = x(x − 5) → x = 0 or x = 5
Check x = 0: √1 = −1 → False (extraneous). Check x = 5: √16 = 4 = 5−1 → True.
x = 5
Rearrange: 0 = x² − 5x
Factor: 0 = x(x − 5) → x = 0 or x = 5
Check x = 0: √1 = −1 → False (extraneous). Check x = 5: √16 = 4 = 5−1 → True.
x = 5
SAT Math
Problem Solving & Data Analysis
This domain tests quantitative literacy — the ability to use ratios, percentages, statistics, and data from real-world contexts. About 29% of SAT Math, all in the calculator section.
Key Concepts
- Ratios and proportions: a/b = c/d, cross-multiply to solve
- Percentage change: (new − old) / old × 100%
- Unit conversion: chain conversion factors to cancel units
- Mean, median, mode, range, and standard deviation
- Interpreting scatterplots and lines of best fit
- Reading two-way frequency tables
- Probability: favorable outcomes / total outcomes
Test Strategy
- Always re-read what the question is actually asking for
- Identify whether context calls for mean or median
- Distinguish "percentage change" from "percentage point change"
- For proportional reasoning, label units to avoid inversion errors
- Look at axis labels and units before reading scatterplot values
Common Mistakes
- Confusing median (middle value) with mean (average) — outliers affect the mean, not the median
- Percentage vs. percentage point: if a rate rises from 20% to 25%, that is a 5 percentage point increase but a 25% relative increase
- Setting up ratios upside-down (inverting numerator and denominator)
- Misreading the scale on compressed or non-zero-based graphs
Practice Problem 1
A car travels 150 miles on 5 gallons of fuel. At this rate, how many gallons are needed to travel 390 miles?
Show Answer & Solution
Rate: 150 mi / 5 gal = 30 mi/gal
Gallons needed: 390 mi ÷ 30 mi/gal = 13 gallons
Gallons needed: 390 mi ÷ 30 mi/gal = 13 gallons
Practice Problem 2
A dataset has the values: 4, 7, 7, 9, 13, 15, 20. A new value of 100 is added. Which measure of central tendency changes the most?
Show Answer & Solution
Original mean = 75/7 ≈ 10.7. New mean = 175/8 ≈ 21.9 (increase of ~11.2).
Original median = 9 (middle of 7 values). New median = (9+13)/2 = 11 (middle of 8 values).
The mean changes dramatically because it is sensitive to extreme outliers. The median shifts only slightly.
Original median = 9 (middle of 7 values). New median = (9+13)/2 = 11 (middle of 8 values).
The mean changes dramatically because it is sensitive to extreme outliers. The median shifts only slightly.
Practice Problem 3
A product's price increased from $80 to $104. By what percentage did the price increase?
Show Answer & Solution
Percentage change = (104 − 80) / 80 × 100% = 24/80 × 100% = 30%
SAT Reading & Writing
Reading & Writing Strategy
The digital SAT combines Reading and Writing into one section: 54 questions in 64 minutes across two modules. Questions cluster around information and ideas, craft and structure, and expression of ideas.
Core Strategies
- Main idea first: before answering specific questions, identify the passage's central claim in one sentence
- Evidence-based answering: every correct answer has direct textual support — eliminate answers that require assumptions
- Vocab in context: use the surrounding sentence, not the word's most common meaning, to determine meaning
- Transition words: identify logical relationships (contrast, cause-effect, continuation) to choose connective words
- Sentence structure: identify the grammatical role each clause plays before choosing punctuation or structure
Timing & Pacing
- Digital SAT: approximately 1 min 10 sec per question average
- Flag and skip difficult questions — return with remaining time
- Read the question stem before the passage on evidence questions
- For rhetorical synthesis questions, identify the goal of the sentence first
Practice Problem 1 — Vocabulary in Context
As used in the sentence "The scientist's tenuous argument failed to persuade even her supporters," the word tenuous most nearly means:
(A) long (B) weak (C) bold (D) detailed
(A) long (B) weak (C) bold (D) detailed
Show Answer & Solution
The context clue is "failed to persuade even her supporters" — this implies the argument lacked convincing force. Tenuous means thin, flimsy, or lacking substance. Answer: (B) weak
Practice Problem 2 — Transition Words
"Renewable energy production has increased significantly over the past decade. ______, fossil fuels still account for more than 80% of global energy consumption."
Which transition word best completes the sentence?
(A) Therefore (B) Additionally (C) Nevertheless (D) Similarly
Which transition word best completes the sentence?
(A) Therefore (B) Additionally (C) Nevertheless (D) Similarly
Show Answer & Solution
The two sentences are in contrast: renewables grew, yet fossil fuels still dominate. A contrast transition is needed. Nevertheless signals contrast or concession. Answer: (C) Nevertheless
ACT Prep
ACT-Specific Differences
The ACT and SAT test similar content, but the format and pacing are meaningfully different. Understanding those differences lets you allocate preparation time more effectively.
| Feature | SAT (Digital) | ACT |
|---|---|---|
| Score Range | 400–1600 | 1–36 composite |
| Sections | Reading & Writing, Math | English, Math, Reading, Science |
| Science Section | No (data analysis in math) | Yes — 35 min, 40 questions |
| Pacing | More time per question | Faster — especially English and Science |
| Wrong Answer Penalty | No penalty | No penalty — always guess |
| Calculator | Digital (Desmos integrated) | Own calculator, entire math section |
| Trigonometry | Minimal | Explicit trig section in math |
ACT Science Strategy
- The Science section tests data interpretation, not science facts — you are reading graphs and tables
- Read the question before reading the passage for efficiency
- Most answers are directly in the figure — avoid over-reading
- Know how to read a standard deviation bar and explain a conflicting viewpoints passage
Taking Both Tests
- Take both official practice tests early to identify which format favors your strengths
- Strong on science? ACT Science section may boost your composite
- Faster reader? ACT pacing is less forgiving but rewards speed
- Most colleges accept both equally — submit whichever score is stronger
Test Strategy
Time Management & Test-Day Strategy
Technical knowledge alone does not guarantee a top score. The students who improve most combine content mastery with a systematic approach to time, question selection, and pressure management.
Pacing by Section (SAT)
- Reading & Writing Module: ~1:10 per question
- Math (no calculator): ~1:35 per question
- Math (calculator): ~1:45 per question
- Flag any question taking more than 2 min — come back
- Leave 3–5 min to review flagged items
Pacing by Section (ACT)
- English: ~30 sec per question (75 Q / 45 min)
- Math: ~1 min per question (60 Q / 60 min)
- Reading: ~8–9 min per passage (40 Q / 35 min)
- Science: ~5 min per passage (40 Q / 35 min)
- Guess on any question you skip — no penalty
Process of Elimination
- Eliminate one or two obviously wrong answers before committing
- On reading questions: eliminate answers with words like "always" or "never" that overstate the passage
- On math: eliminate answers outside a reasonable range before computing
- When down to two answers: re-read the question stem exactly, not your memory of it
Practice Test Schedule
- 8+ weeks out: take a full timed diagnostic to establish baseline
- 6 weeks out: take a second full practice test after targeted review
- 3 weeks out: take a third test, focus review on lingering weak areas
- 1 week out: review errors only — no new content. Prioritize sleep
- Test week: trust your preparation. Maintain sleep schedule. Arrive early
Recommended Books
SAT / ACT Textbooks
These are the books Dr. Preston recommends to students. Amazon links use the fissionlab-20 affiliate tag — your purchase supports free content at no additional cost to you.
The Official SAT Study Guide
College Board — The only official source
Eight full-length official practice tests with answer explanations. The gold standard for SAT prep — use real questions whenever possible.
View on Amazon →Cracking the SAT
Princeton Review
Strong strategy coverage with process-of-elimination techniques, pacing drills, and section-specific walkthroughs. Good complement to official practice.
View on Amazon →Barron's SAT
Barron's Educational Series
Comprehensive subject review with additional practice tests. Particularly strong on vocabulary in context and grammar rules for the Writing section.
View on Amazon →PWN the SAT Math Guide
Mike McClenathan
The best dedicated SAT Math resource available. Rigorous problem breakdowns and a mastery-based approach. Recommended for students targeting 750+ on Math.
View on Amazon →Join the Free Study Community
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