SAT Math Exhaustive Study Guide and Strategy Notes

SAT Math Test Structure and Content

• Test Composition and Scoring

  • The test is divided into modules, with varying levels of difficulty (easier and harder), determined by performance in the first module.
  • Each module contains two pretest questions that do not impact the student's score. These are unmarked, making them indistinguishable from graded questions.
  • Content Distribution:
    • Algebra: $35\%$ (approximately $7$ to $8$ questions per module).
    • Advanced Math: $35\%$ (approximately $7$ to $8$ questions per module).
    • Problem Solving: $15\%$ (approximately $3$ to $4$ questions per module).
    • Trig and Geometry: $15\%$ (approximately $3$ to $4$ questions per module).

• Math Topic Coverage

  • Elementary School Math: Included.
  • Algebra 1 and Algebra 2: Included.
  • Geometry: Included.
  • Trigonometry: Included.
  • Statistics: Included, but limited to the introductory level (equivalent to the first two to three weeks of an AP Stats course), not full AP level.
  • Pre-Calculus and Calculus: Not included.

• Question Types and Formats

  • Slicing by Skill: Problem Solving constitutes approximately $70\%$ ($15$ to $16$ questions per module) while Word Problems make up $30\%$ ($6$ to $7$ questions per module).
  • Slicing by Format:
    • Multiple Choice: $75\%$ ($16$ to $17$ questions).
    • Fill-in: $25\%$ ($5$ to $6$ questions).

Problem Solving Strategies and Pacing

• Pacing Hierarchy

  • Perform questions that can be answered quickly and accurately first.
  • Prioritize questions where Princeton Review strategies are applicable.
  • Pacing Tools:
    • Answer and Advance: For confident answers.
    • Guess and Go: Used for quick guessing when time is short.
    • Mark and Move: Flagging a question to return to later.
    • Skip for Now: Moving past a difficult item to secure easier points.

• Guessing Strategy (Multiple Choice vs. Fill-in)

  • If a student has one multiple choice and one fill-in question remaining with time to solve only one, they should solve the fill-in.
  • Probability Logic: Multiple choice offers a one-in-four chance ($25\%$) of guessing correctly. Fill-in questions have a near-zero probability of a correct guess. Increasing the probability of the fill-in through work while guessing on the multiple choice is mathematically superior.

• Scratch Paper Habits

  • Always write work down to prevent careless errors that occur when calculating in the head.
  • Setup the math clearly (e.g., sections a, b, c) to track operations.
  • Geometry: Draw the figure even if one is provided in the test software. The provided figure may not contain all the details mentioned in the text. Drawing your own allows you to label every piece of information provided.

• The Trap of the Final Question

  • The SAT often designs problems where a common intermediate variable (like $a$) is solved for, but the question actually asks for a different variable (like $b$).
  • RTFQ (Read The Final Question): Read the final question before solving, and again after marking the answer to ensure the correct value is being submitted.
  • The Highlighter Tool: Note that the highlighter tool for math has been removed by the College Board, making it even more important to note keywords on scratch paper.

Word Problem Methodology

• Basic Approach to Word Problems

  1. RTFQ (Read the Final Question): Identify exactly what the question is asking for first.
  2. Let the Answers Point the Way: Use the options to understand the scale and format required.
  3. Work in Bite-Sized Pieces: Solve one small part of the problem at a time rather than trying to digest the whole text at once.
  4. Use Process of Elimination (POE): Eliminate choices after each intermediate step to narrow down the field.

• Case Study: The Runner and Average Rate

  • Problem: A runner is in a $10\,\text{km}$ race. After $32.5\,\text{minutes}$, they are $3.5\,\text{km}$ from the finish.
  • Strategy: Draw a picture. Start at $0$, finish at $10$. The runner is at $6.5\,\text{km}$ ($10 - 3.5 = 6.5$).
  • Rate Calculation: $\text{Rate} = \frac{\text{Distance}}{\text{Time}} = \frac{6.5\,\text{km}}{32.5\,\text{min}}$.
  • Ballparking: Approximate $\frac{6.5}{32.5}$ as $\frac{6}{30}$, which is $\frac{1}{5}$ or $0.2$. This allows for quick elimination of answers that are too large or too small.

• Case Study: Fruit Purchase

  • Problem: Total fruit is $360$ pieces. $1/5$ are apples ($72$), $1/3$ are oranges ($120$). A quarter of the remaining are pears. How many are strawberries?
  • Step-by-Step:
    1. Apples $+$ Oranges $= 72 + 120 = 192$.
    2. Remaining fruit $= 360 - 192 = 168$.
    3. If pears are $1/4$ of the remaining, then strawberries are $3/4$ of the remaining ($168$).
    4. $3/4$ of $168$ is $126$. Elimination can be used to see that $126$ is much more likely than lower numbers like $90$.

Advanced Techniques: Elimination and Ballparking

• Literal and Equivalent Expressions

  • To find an equivalent expression, focus on a single piece of the expression, such as the highest-degree term (e.g., $n^3$) or the constant term.
  • Example: For $(8n^3 + \dots) + (21n^3 + \dots)$, the sum must contain $29n^3$. Eliminate any option that does not.
  • Then check the constant: $16 - 27 = -11$. If the constant is negative, eliminate options with positive constants. This avoids expanding the entire polynomial.

• Roman Numeral Questions

  • Start with the statement that appears in exactly half of the answer choices. This maximizes the number of answers eliminated regardless of whether the statement is true or false.
  • Alternatively, start with the "easiest" number to test.
  • Once a statement is proven false, eliminate all options containing that numeral.

• Ballparking Logic

  • Eliminate answers that are logically impossible (too big or too small).
  • Example: If a problem asks for $75\%$ of $120,000$, the answer must be greater than half ($60,000$) but less than the total. If only one choice fits that range, no calculation is needed.
  • Unit Conversions: When converting $8,000\,\text{milliliters}$ to liters ($1000\,\text{ml} = 1\,\text{L}$), you are looking for a number smaller than $8,000$. Set up a ratio: $\frac{1,000\,\text{ml}}{1\,\text{L}} = \frac{8,000\,\text{ml}}{x}$. Cross-multiply to find $1,000x = 8,000$, yielding $x = 8$.

Plug In The Answer (PETA)

• PETA Concept (The "Pita Bread" Strategy)

  • Use when the question asks for a specific value and the answer choices are numbers.
  • Instead of solving algebra, plug the answer choices back into the original equation.

• PETA Steps:

  1. Rewrite and Label: List the answer choices on scratch paper and label what they represent.
  2. Start in the Middle: Begin with one of the middle numerical values (Choice B or C). This allows you to see if you need a larger or smaller number.
  3. Bite-Sized Pieces: Calculate parts of the equation separately.
  4. Eliminate and Stop: When an answer works, select it and move on.

• Example: Rational Equation

  • Equation: $\frac{20}{a-1} - \frac{18}{a+1} = 2$.
  • Plugging in $a = 4$: $\frac{20}{3} - \frac{18}{5}$. This yields a complex fraction, not $2$.
  • Plugging in $a = 5$ (Middle Choice): $\frac{20}{4} - \frac{18}{6} = 5 - 3 = 2$. This works perfectly.

• Example: System of Equations

  • Equation 1: $y + 4 = x$. Equation 2: $-\frac{2}{3}y = 1 + x$.
  • Plug the coordinates $(x, y)$ from the choices into the simpler equation first to eliminate distractors before moving to the more complex equation.

Digital SAT Rules and Tools

• Fill-In Question Rules

  • Positive Answers: Can enter up to $5$ characters (e.g., $0.333$ or $1/3$).
  • Negative Answers: Can enter up to $6$ characters (including the negative sign).
  • Long Decimals: Enter as much of the decimal as will fit. If the answer is $1/3$, enter $0.333$ (using all five positions).
  • Fractions: Can be reduced or unreduced, as long as they fit in the designated character space. Mixed numbers must be entered as improper fractions or decimals.

• Calculator and Reference Features

  • Desmos: The built-in graphing calculator can solve equations, find intercepts, and locate minima/maxima by hovering over points. It can also convert results to fractions with a single button click.
  • TI-84 Tips: Use the "Math" key to help with fraction reduction if not using Desmos.
  • Reference Table: Available within the software for common formulas (volumes, areas).

Questions & Discussion

• The Microphone Effect: A psychological phenomenon described during the session where a student's IQ metaphorically "drops to their shoe size" because their microphone is on and they are on the spot. This highlights why it is helpful for classmates to help each other.
• Interaction with Arush/Harish: Discussion on focusing on specific terms like $n^3$ to simplify polynomial equivalence problems.
• Interaction with Anastasia and Gabrielle: Discussion on Roman Numeral strategy. Gabrielle noted starting with information that appears most frequently to aid elimination.
• Interaction with Dennis: Breakdown of the fruit word problem, emphasizing the importance of the word "remaining."
• Interaction with Dylan and Mackenzie: Visualizing the runner's path to determine actual distance traveled vs. distance remaining.
• Interaction with Samuel: Detailed breakdown of the Ferris wheel vs. Eiffel Tower problem ($Total = 1,248$; $Eiffel = Ferris + 720$). Samuel correctly noted using ballparking to eliminate overly large Ferris wheel heights instantly.
• Interaction with Noah: Solving a radical equation ($a - 5 = -7 + \sqrt{3a + 10}$) through PETA, highlighting that the value under the square root must result in a real number and the whole side must be positive.
• Interaction with Rita: Analyzing data tables and realizing that when $x$ values are identical across tables, you only need to plug in one specific $x$ (like $0$) to check which table's $y$ value satisfies the inequality.

Homework and Final Reminders

• RTFQ Again: Read Keywords. Write them on scratch paper.
• Drills: Complete the assigned drill in the "Lessons" section of the coursework tab. Submit answers online for full explanations.
• Self-Study: Chapters labeled "101" are intended for self-study and independent review.
• Exit Procedure: There is no "exit" button in the software interface; users must close the browser tab to leave the session.