The Ultimate ACT Math Cram Sheet: Advanced Concepts Edition

Exam Overview & Format

The ACT Math test is a sprint, not a marathon. It is designed to test your ability to work quickly and accurately while switching gears between different areas of math conceptually. Unlike the SAT, the ACT allows a calculator for the entire math section.

Calculators Allowed: Most TI-83/84 models, Casio fx and scientific calculators.
Calculators BANNED: TI-89, TI-92, TI-Nspire CAS (any CAS models), calculators with QWERTY keypads.

Scoring & What You Need

The ACT is scored on a scale of 1-36. The Math score constitutes 25% of your Composite Score.

The "Curve"

Because there are 60 questions, the margin for error at the top is incredibly slim. Here is a rough estimate based on recent test forms:

• 36 (Perfect): 59–60 correct
• 34-35 (Elite): 56–58 correct
• 30 (Advanced): ~50–52 correct
• 20 (National Average): ~30–32 correct

Penalties

• No Guessing Penalty: Never leave a bubble blank. If you have 30 seconds left and 5 empty questions, pick a "Letter of the Day" (e.g., all B/G) and bubble them in.

Strategy by Difficulty Band

Since the Math section gets progressively harder, your strategy must evolve as you move through the test.

Questions 1–20 (The Sprint)

• Goal: 100% accuracy in ~12–15 minutes.
• Strategy: These are foundational Pre-Algebra and basic Geometry concepts. Don't overthink them. If you are stuck here for more than 45 seconds, you are likely missing a simple shortcut or misreading the question.
• Time Check: If you aren't at Question 20 by the 15-minute mark, you need to speed up to have time for the complex concepts at the end.

Questions 21–40 (The Grind)

• Goal: Accuracy with speed. Complete by minute 35–40.
• Strategy: This is where word problems and multi-step algebra kick in. Use PITA (Plug In The Answer). If a question asks "What is the value of x?" and gives you numbers, plug answer choice C (or H) into the equation first to see if it works.

Questions 41–60 (The Advanced Zone)

• Goal: Survive and scavenge points. Spend your remaining time here.
• Strategy: This section contains the "Advanced Concepts" (Logarithms, Matrices, Vectors, Ellipses).
• The "Eye-Ball" Method: Diagrams are usually drawn to scale in the ACT (unless stated "Not drawn to scale"). If you forgot the geometry formula, estimate lengths or angles visually to eliminate absurd answer choices.
• Pick Numbers: If variables are in the question and the answer choices, substitute simple integers (avoid 0 and 1) for the variables to solve.

Highest-Yield Content Review

This section skips basic algebra and focuses entirely on the advanced topics that appear in the final 20 questions. These are the concepts that separate a 28 from a 34.

1. Complex Numbers

• Definition: $i = \sqrt{-1}$ and $i^2 = -1$.
• The Cycle: $i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$.
• Strategy: For high powers like $i^{53}$, divide the exponent by 4. The remainder determines the value. ($53 \div 4 = 13$ with remainder 1, so $i^{53} = i^1 = i$).
• Conjugates: To rationalize a denominator $\frac{a}{b+ci}$, multiply top and bottom by the conjugate $(b-ci)$.

2. Logarithms

• Definition: $\log_b(x) = y$ is the same as $b^y = x$.
• Crucial Rules:
  • Product Rule: $\log<em>b(xy) = \log</em>b(x) + \log_b(y)$
  • Quotient Rule: $\log<em>b(\frac{x}{y}) = \log</em>b(x) - \log_b(y)$
  • Power Rule: $\log<em>b(x^p) = p \cdot \log</em>b(x)$
  • Change of Base: $\log_b(a) = \frac{\log(a)}{\log(b)}$ (Use specific vals on calculator)

3. Matrices

• Addition/Subtraction: Only possible if dimensions match. Add corresponding elements.
• Multiplication: Possible if Columns of A = Rows of B. Multiply Row by Column.
• Determinant: For a $2 \times 2$ matrix $\begin{bmatrix} a &amp; b \ c &amp; d \end{bmatrix}$, the determinant is $ad - bc$.
  • If determinant is 0, the matrix has no inverse.

4. Vectors

• Component Form: $v = \langle x, y \rangle$. Often written as $ai + bj$.
• Addition: Add X components together and Y components together: $\langle 3, 4 \rangle + \langle 1, -2 \rangle = \langle 4, 2 \rangle$.
• Magnitude: The length of the vector using Pythagorean theorem: $||v|| = \sqrt{x^2 + y^2}$.

5. Advanced Geometry & Conics

6. Trigonometry (SOH CAH TOA + Advanced)

• Identities:
  • $\sin^2 \theta + \cos^2 \theta = 1$
  • $\tan \theta = \frac{\sin \theta}{\cos \theta}$
• Law of Sines: $\frac{\sin A}{a} = \frac{\sin B}{b} = \frac{\sin C}{c}$ (Use for non-right triangles when you have a matching angle-side pair).
• Law of Cosines: $c^2 = a^2 + b^2 - 2ab\cos C$ (Use for SAS or SSS triangles).
• Period of a function: For $y = A\sin(Bx)$, the Period is $\frac{2\pi}{B}$ (or $\frac{360^{\circ}}{B}$).

7. Sequences & Series

8. Probability & Statistics

• Expected Value: Multiply each outcome by its probability and sum them up. $E(x) = P<em>1 \cdot X</em>1 + P<em>2 \cdot X</em>2 + …$
• Permutations (Order Matters): $nPr$. Putting people in seats, electing officers.
• Combinations (Order Doesn't Matter): $nCr$. Picking a team of 3 people from a group of 10.

Common Pitfalls & Traps

1. Units Mismatches: The question gives speed in miles per hour but asks for distance in feet or time in minutes. ALWAYS check the units in the answer choices first.
2. "Integers" vs. "Real Numbers":
   • If it says "integers," fractions and decimals are illegal. Negative numbers and zero are legal.
   • If it says "real numbers," everything is fair game.
3. Forgetting $\pm$: When solving $x^2 = 25$, the answer is $x = 5$ AND $x = -5$. The ACT loves to trap students who forget the negative root.
4. Percent Change: The formula is $\frac{\text{New} - \text{Old}}{\text{Old}} \times 100$. Don't divide by the New number.
5. Mean vs. Median: The Mean is the average (sensitive to outliers). The Median is the middle number (resistant to outliers).
6. Excluded Values: In rational equations like $\frac{x^2-9}{x-3}$, $x$ cannot equal 3 (division by zero). Watch for "undefined" traps.
7. Degrees vs. Radians: Check your calculator mode! If you are plugging in $\sin(30)$ and expecting 0.5 but get -0.988, you are in Radian mode. Default to Degree mode unless the problem explicitly uses $\pi$.

Memory Aids & Mnemonics

Important Dates (2024-2025 Cycle)

Note: Dates are approximate based on standard ACT scheduling. Always verify on ACT.org.

Last-Minute Tips & Test Day Checklist

The Night Before

• Battery Check: Put fresh batteries in your calculator. Bring a backup set (or a backup calculator).
• Pack the Bag: Printed admission ticket, photo ID, #2 pencils (wooden only, NO mechanical), calculator, erasing shield/good eraser, water, and a high-protein snack.
• Stop Studying: Do not do practice problems after 8 PM. Your brain needs to rest to process speed tomorrow.

Test Day Mental Prep

• Warm Up: Solve 3-5 simple math problems (like $15 \times 12$ or a simple equation) before you walk into the test center to wake up your "math brain."
• Pacing Reset: The Math section is Section 2. Shake off the English section (Section 1) completely. It doesn't matter anymore.
• The "Letter of the Day": Pick one letter (e.g., B/G) before the test starts. If you run out of time on the last 5 questions, bubble that letter immediately without reading. Do not zigzag your guesses.

Final Encouragement: You don't need to be a mathematician to beat this test; you just need to be a strategic test-taker. Trust your prep, watch the clock, and hunt for the easy points. You've got this!