Comprehensive SAT Prep Notes: English Grammar & SAT Math Formulas

General Grammar Rules

• Semicolons: always come before conjunctive adverbs. Example: "elephants are large creatures; however, they are extremely gentle."
• Colons: can replace periods if the second sentence further elaborates the first one. Example: "Vani worked on her science fair project all year: the work tests the detriments of cancer cells."
• Filler words before colons are incorrect (colons should introduce elaboration, not serving as fillers).
  • Incorrect: "She looked for many ingredients, such as milk, carrots, and flour."
  • Correct usage avoids fillers before colons.
• Transitional phrases: apply transitional phrases that logically connect two clauses, especially when negating or contrasting ideas.
  • Be aware of contradicted or reversed phrases when negation is involved.
• Commas: used to separate adjectives whose order is interchangeable in a sentence.
  • Example: "Mr. Otto is a cool, smart teacher." (adjectives of equal rank)
• Dangling modifiers: place descriptive phrases next to the noun they modify; fix with an explicit subject.
  • Incorrect: "Running down the street, the backpack fell."
  • Correct: "Running down the street, I dropped my backpack."
• Faulty comparisons: avoid making unclear or illogical comparisons; ensure subjects being compared are parallel and make sense.
  • Example (garbled in transcript): "Vani’s videos were as famous as Fatimah vani’s videos were as famous as Fatiman’s." (illustrates avoidable redundancy and unclear comparison)
• Quantifiers and number agreement:
  • plural vs. singular cues: "number, many, fewer" go with plural nouns;
  • "amount, much, less" go with singular nouns.
• Vocabulary roots (common affixes and roots)
  • duc/duct: lead
  • dict: say/speak
  • Scrib/script: write
  • Struct: build
  • vid/vis: see
  • port: carry
  • man: hand
  • phil: love
  • path: feeling
  • cred: believe
  • tract: pull
  • rupt: break
  • spec: look/see closely
  • sta/sti: stand
  • auto: self
  • clam: shout
  • mis-: wring/incorrectly
  • a-: not
  • dox: opinion
• Examples of vocabulary from roots (illustrative):
  • induce, conduct
  • verdict, dictate
  • inscription
  • infrastructure
  • envision, evident
  • deport, portable
  • manipulate
  • epilogue, logic
  • bibliophile
  • apathetic
  • credence
• Memorize meaning of word families and how roots hint at meanings.

Test-Taking Strategies (Grammar)

• If all answer options are grammatically correct, choose the most concise one.
• Watch transition words and how sentences connect.
• Read the rule or guideline implied by the grammar question.
• Check the relationship between sentences; ensure coherence across the full sentence.
• Choose answers that sound natural; trust intuition about what sounds right.
• Time pacing: start with easier grammar questions, then loop back to reordering or harder items.
• Eliminate options that violate independent/dependent clause rules or create ambiguity.
• Be mindful of common grammar rules involving clauses, punctuation, and agreement.

SAT Math Formulas: Algebra & Functions

• Point-slope form:
  • y - y1 = migl(x - x1igr)
• Standard form of a line:
  • Ax + By = C
• Quadratic formula (solve ax^2 + bx + c = 0):
  • x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
• Discriminant (to determine number of real solutions):
  • \Delta = b^2 - 4ac
  • If \Delta > 0 → 2 real solutions; if \Delta = 0 → 1 real solution; if \Delta < 0 → 0 real solutions.
• Sum and product of roots for ax^2 + bx + c = 0:
  • Sum of roots: -\frac{b}{a}
  • Product of roots: \frac{c}{a}
• Exponent rules (basic familiarity): e.g., a^m a^n = a^{m+n}, (a^m)^n = a^{mn}, a^0 = 1.
• Polynomial identities:
  • Difference of squares: a^2 - b^2 = (a - b)(a + b)
  • Perfect square trinomial: a^2 \pm 2ab + b^2 = (a \pm b)^2

Percent, Ratios, and Proportions

• Percent change: \text{percent change} = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%
• Part/Whole percent: part/whole × 100% (apply to problems where a portion of a total is sought).
• Ratio: expressed as part-to-part or part-to-whole; used to compare quantities.

Rates, Distance, and Work Problems

• Distance, rate, time: d = r\,t
• Work problems (combined work): if two workers with times T1 and T2 work together, their combined rate is 1/T1 + 1/T2, and total time T satisfies:
  • \frac{1}{T} = \frac{1}{T1} + \frac{1}{T2}
• Direct variation: y \propto x \;\Rightarrow\; y = kx
• Inverse variation: y \propto \frac{1}{x} \;\Rightarrow\; y = \frac{k}{x}

Circles and Geometry (SAT-focused): Basics

• Circle equation:
  • (x - h)^2 + (y - k)^2 = r^2 where center is $(h,k)$ and radius is $r$.
• Radius and diameter relations:
  • Diameter = 2 × radius.
• Unit conversions and circle measures:
  • Radians to degrees: 360^{\circ} = 2\pi \text{ radians}
  • Arc length (with angle in radians): s = r\theta
  • Sector area (angle in radians): A = \frac{1}{2} r^2 \theta
  • Sector area (angle in degrees): A = \frac{\theta}{360} \pi r^2

Circle Geometry and Trigonometry (named terms)

• Terms: tangent, segment, chord, secant (definitions and typical use in problems).
• Common circle-related problem types include calculating arc length, sector area, chord length, and radius/diameter relationships.

Systems of Equations and Inequalities

• Methods: elimination, substitution, graphing.
• Intersection point: the solution to the system.
• Inequalities: flip the inequality sign when multiplying/dividing by a negative number.
• Systems with multiple equations can be split or solved as a combined set to find solutions.

Statistics & Probability

• Mean (average): \text{mean} = \frac{\sum x_i}{n}
• Median: the middle value in an ordered list; if even number of terms, average the two middle values.
• Mode: the most frequently occurring value.
• Range: \text{range} = \text{max}(x) - \text{min}(x)
• Probability: P(A) = \frac{\text{number of favorable outcomes}}{\text{total number of possible outcomes}}
• Other concepts to study: basic probability rules, counting principles, and interpreting data visuals.

Trigonometry and Geometry Notions (SAT-minimum basics)

• Tangent, secant, chord relations, and circle segment properties.
• Remember to relate geometric figures to equations and algebraic representations.

Test-Taking and Calculator/Technology Tips

• Desmos tips (graphing calculator):
  • Use square roots and cube roots: \sqrt{ }, \sqrt[3]{ }.
  • For circle-related tasks, use diameter and radius references in graphs.
  • Arc and section (arc) utilities for circle problems.
• Problem-solving workflow tips:
  • If you can do a problem in about 230 seconds, plug in values and check feasibility before expanding fully.
  • When possible, solve without multiplying out (keep expressions factored or in simple forms).
  • Check if answers can be tested by substituting into simpler equations or into approximate values.
  • Use a table or regression tools in Desmos for linear fits when relevant.
  • For systems, you can plug in equations for quick checks; median and mean can be used conceptually in some data-based prompts.
• Common pitfalls to avoid:
  • Avoid non-simplified equations when a simpler form is available.
  • Be cautious with non-unique or no-solution systems; not all systems have a real intersection.
  • Ensure correct handling of vertical lines (x = constant) in coordinate geometry problems.
• Practice approach for “no solution” or inconsistent systems: identify when equations contradict each other and confirm with a quick check.

Quick Reference: Examples and Mini-Essentials

• Point-slope form sample: y - y1 = m\,(x - x1)
• Standard form and discriminant reminder: \Delta = b^2 - 4ac with solution counts based on sign of \Delta.
• Circle standard form reminder: (x - h)^2 + (y - k)^2 = r^2
• Sector area and arc length basics: A = \tfrac{1}{2}r^2\theta, \quad s = r\theta\quad (\theta\text{ in radians})
• Percent change example form: \%\Delta = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%
• Distance-rate-time shorthand: d = r t
• Direct vs. inverse variation forms:
  • Direct: y = kx
  • Inverse: y = \dfrac{k}{x}

Connections to Foundational Principles

• Grammar rules reinforce sentence structure, coherence, and clarity—key for effective communication in math explanations and word problems.
• Algebra and functions build the language of quantitative reasoning; mastering forms (slope, standard form, discriminant) supports problem solving across disciplines.
• Geometry ties algebra to shapes; understanding circle equations and area/arc relationships enables solving geometry problems with algebraic methods.
• Data interpretation in statistics underpins decision-making in real-world contexts and informs probability reasoning.

Practical and Ethical Implications (study-oriented)

• Precision in notation (using exact LaTeX forms) improves clarity and reduces misinterpretation in mathematical communication.
• Ethical use of calculators and tools: use Desmos andTech appropriately for learning, not for cheating; understand steps behind results.
• Critical thinking: distinguish between well-posed problems and ambiguous prompts; ask clarifying questions if needed in a real exam setting.

Summary of Key Formulas and Concepts (Condensed)

• Point-slope: y - y1 = m\,(x - x1)
• Standard line: Ax + By = C
• Quadratic solution: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
• Discriminant: \Delta = b^2 - 4ac
• Sum of roots: -\frac{b}{a}; Product of roots: \frac{c}{a}
• Circle: (x - h)^2 + (y - k)^2 = r^2
• Sector area: A = \tfrac{1}{2} r^2 \theta \text{(radians)}
• Arc length: s = r\theta \text{(radians)}
• Percent change: \%\Delta = \frac{\text{new} - \text{old}}{\text{old}} \times 100\%
• Distance: d = r t
• Direct: y = kx; Inverse: y = \dfrac{k}{x}
• Population/mean/median/mode/range basics: mean = sum/n; median = middle value; mode = most frequent; range = max - min
• Probability: P(A) = \frac{\text{favorable}}{\text{total}}