SAT (Digital) Math Formulas & Equations — Night-Before Cram Sheet
1. Exam Overview & Format (REQUIRED)
You’re taking the Digital SAT (Bluebook app). It’s adaptive by section (Module 1 → Module 2).
Structure (Digital SAT)
| Section | Modules | # Questions | Time | Question types | % of total score |
|---|---|---|---|---|---|
| Reading & Writing | 2 | 54 (27 + 27) | 64 min (32 + 32) | Multiple-choice | 50% |
| Math | 2 | 44 (22 + 22) | 70 min (35 + 35) | Multiple-choice + Student-Produced Response (SPR) | 50% |
Total testing time: 134 min (2h 14m)
Breaks: Typically one 10-minute break between Reading/Writing and Math.
Calculator / Reference Sheet / Materials (high-yield)
- Calculator: Allowed for all Math on the Digital SAT.
- Bluebook includes a built-in graphing calculator (Desmos-style).
- You may also bring an approved handheld calculator.
- Math reference sheet: Provided in-test (basic geometry + a few constants). Don’t rely on it for algebra—know your core formulas.
- Scratch work: Test center provides scratch paper/notes materials (policies can vary slightly by center).
Critical: Because the test is adaptive, Module 1 matters a lot. Don’t “save energy” early—bank points immediately.
2. Scoring & What You Need (REQUIRED)
How scoring works
- Total score: 400–1600
- Section scores: Reading & Writing 200–800, Math 200–800
- Digital SAT uses adaptive testing: your Module 2 difficulty depends partly on Module 1 performance.
- Scores are scaled/equated, not a simple “# correct × points.”
Guessing / penalties
- No penalty for guessing. If you can eliminate even 1 choice, guess aggressively.
“What score do you need?” (practical targets)
- There’s no passing score. Targets depend on your colleges/scholarships.
- Common goalposts students use:
- 1200–1300: solid baseline for many schools
- 1400+: competitive at many selective schools
- 1500+: top-tier range
Score release timing (non-fabricated guidance)
- Score release timing can vary by test date and administration. Check your College Board account for the official release date for your test.
3. Section-by-Section Strategy (REQUIRED)
Reading & Writing (2 modules, 64 minutes)
- Don’t get stuck. If a passage/question is sticky after ~45–60 seconds, guess, flag, move on.
- Use “line evidence” discipline. Every correct answer must be supported by words in the text—avoid “sounds right.”
- For grammar: read the sentence without the underlined portion first; then pick the option that is clear + grammatically correct.
- Module 1 matters: treat it like it’s worth extra—higher performance can unlock a higher-ceiling Module 2.
Math (2 modules, 70 minutes) — where formulas/equations win
Time budget: 44 questions / 70 min ≈ 1 min 35 sec per question.
- Two-pass method (per module):
- Pass 1: anything you can do in ≤60–75s.
- Pass 2: return for algebra grind/geometry setup.
- Exploit the built-in graphing calculator strategically:
- Use it to check or to solve intersection points / zeros / systems.
- But don’t overuse it for simple algebra—you’ll waste time typing.
- SPR (student-produced response) rules:
- You enter the number (can be integer, decimal, or fraction).
- Don’t round unless told. If the exact value is nice (like $\frac{3}{4}$ or $\sqrt{2}$ is not allowed typically), give exact decimal/fraction as appropriate.
- Watch negatives and parentheses.
- When choices are numeric, backsolve:
- Plug answer choices into the equation (start with middle choice) to avoid heavy algebra.
- Always do a “sanity check”:
- Sign? Units? Does the answer magnitude make sense (e.g., probability >1 is impossible)?
Module-adaptive tip: Accuracy beats speed. Fast-but-wrong hurts twice.
4. Highest-Yield Content Review (REQUIRED)
This is the top 20% of formulas/equations that show up constantly—master these.
A) Algebra: linear, systems, inequalities (must-know)
| Topic | High-yield formulas / reminders |
|---|---|
| Slope | m=\frac{y2-y1}{x2-x1} |
| Slope-intercept | y=mx+b (b = y-intercept) |
| Point-slope | y-y1=m(x-x1) |
| Standard form | Ax+By=C (often for intercepts) |
| Midpoint | \left(\frac{x1+x2}{2},\frac{y1+y2}{2}\right) |
| Distance | d=\sqrt{(x2-x1)^2+(y2-y1)^2} |
| Parallel / perpendicular | Parallel: same $m$. Perpendicular: m1m2=-1 |
| Systems (2 lines) | 1 solution: intersect; 0: parallel; infinite: same line |
| Inequalities | Multiply/divide by negative → flip inequality sign |
| Absolute value |
Equation-solving moves that pay off constantly
- Clear fractions early (multiply both sides by LCM).
- Combine like terms; isolate variable.
- If you square both sides, check for extraneous solutions.
B) Exponents, radicals, and algebraic manipulation
| Rule | Formula |
|---|---|
| Exponent product | a^m\cdot a^n=a^{m+n} |
| Exponent quotient | \frac{a^m}{a^n}=a^{m-n} |
| Power of a power | (a^m)^n=a^{mn} |
| Power of product/quotient | (ab)^n=a^n b^n,\quad \left(\frac{a}{b}\right)^n=\frac{a^n}{b^n} |
| Negative exponent | a^{-n}=\frac{1}{a^n} |
| Fraction exponent | a^{1/n}=\sqrt[n]{a},\quad a^{m/n}=\sqrt[n]{a^m} |
| Radical simplify | \sqrt{ab}=\sqrt{a}\sqrt{b} (for $a,b\ge0$) |
| Rationalize | \frac{1}{\sqrt{a}}\to \frac{\sqrt{a}}{a}; for binomials use conjugate |
Common SAT patterns
- Difference of squares: a^2-b^2=(a-b)(a+b)
- Perfect square trinomials: a^2\pm2ab+b^2=(a\pm b)^2
C) Quadratics & polynomials (the SAT staples)
| What you need | Formula / cue |
|---|---|
| Quadratic form | ax^2+bx+c |
| Factoring | Look for two numbers multiply to $ac$ and add to $b$ (or simple $c$ if $a=1$) |
| Quadratic formula | x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} |
| Discriminant | \Delta=b^2-4ac: $\Delta>0$ two real, $\Delta=0$ one real, $\Delta<0$ none real |
| Vertex (x-value) | x=-\frac{b}{2a} |
| Vertex form | y=a(x-h)^2+k vertex $(h,k)$ |
| Zeros / roots | Solve $ax^2+bx+c=0$ (x-intercepts) |
| Sum/product of roots | For ax^2+bx+c=0: r1+r2=-\frac{b}{a},\quad r1r2=\frac{c}{a} |
High-yield equation tricks
- If you’re asked for value at vertex: find $x=-\frac{b}{2a}$, then plug in.
- If you’re asked for maximum/minimum: parabola opens up if $a>0$ (min), opens down if $a<0$ (max).
D) Functions (notation + transformations)
| Concept | One-liner you need |
|---|---|
| Function notation | $f(x)$ is output when input is $x$ |
| Evaluate | Replace $x$ with the given input carefully: $f(2)$ means plug in 2 everywhere $x$ appears |
| Domain | Denominator $\ne 0$; even roots require inside $\ge 0$ |
| Composition | $(f\circ g)(x)=f(g(x))$ |
| Inverse idea | Swap $x,y$ then solve for $y$ (when one-to-one) |
| Transformations | $f(x-h)$ shifts right $h$; $f(x)+k$ shifts up $k$; $-f(x)$ flips over x-axis; $f(-x)$ flips over y-axis |
E) Ratios, rates, percent, and word-equation setup
| Topic | Formula |
|---|---|
| Percent change | \%\text{ change}=\frac{\text{new-old}}{\text{old}}\times 100\% |
| Simple interest | I=Prt |
| Unit rate | \text{rate}=\frac{\text{amount}}{\text{time}} |
| Proportion | \frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc |
| Average | \text{average}=\frac{\text{total}}{#} |
Translation mini-dictionary
- “of” → multiply
- “per” → divide
- “is” → equals
- “more than” → add
- “less than” → subtract (watch order: “5 less than x” = $x-5$)
F) Geometry you actually need (formulas + equation setups)
1) Triangles & right triangles
| Item | Formula / reminder |
|---|---|
| Triangle area | A=\frac{1}{2}bh |
| Pythagorean theorem | a^2+b^2=c^2 |
| 45-45-90 | Legs $x,x$; hypotenuse $x\sqrt{2}$ |
| 30-60-90 | Short $x$; long $x\sqrt{3}$; hypotenuse $2x$ |
| Similar triangles | Corresponding sides proportional; scale factor squares for area |
2) Circles
| Item | Formula |
|---|---|
| Circumference | C=2\pi r=\pi d |
| Area | A=\pi r^2 |
| Arc length | \text{Arc}=\frac{\theta}{360^\circ}\cdot 2\pi r |
| Sector area | \text{Sector}=\frac{\theta}{360^\circ}\cdot \pi r^2 |
| Circle equation (center-radius) | (x-h)^2+(y-k)^2=r^2 |
3) Coordinate geometry (lines + circles show up a lot)
- If they give a graph, you can often read slope/intercepts faster than solving.
- If they give two points: compute slope, then use point-slope form.
4) Area/volume (know the “big 4”)
| Shape | Formula |
|---|---|
| Rectangle | A=lw |
| Parallelogram | A=bh |
| Trapezoid | A=\frac{1}{2}(b1+b2)h |
| Circle | A=\pi r^2 |
| Rectangular prism | V=lwh |
| Cylinder | V=\pi r^2 h |
G) Data, stats, probability (frequent, usually quick points)
| Topic | SAT-level must-knows |
|---|---|
| Mean | \bar{x}=\frac{\text{sum}}{n} |
| Median | Middle value when ordered (or average of two middles) |
| Range | max − min |
| Standard deviation (concept) | Measures spread; larger = more spread out |
| Weighted average | \frac{\sum (w\cdot x)}{\sum w} |
| Probability | P(A)=\frac{\text{favorable}}{\text{total}} |
| “And” (independent) | Multiply probabilities |
| “Or” (mutually exclusive) | Add probabilities |
Data trap to avoid: “Average” is usually mean unless they say median.
5. Common Pitfalls & Traps (REQUIRED)
Sign mistakes in solving equations
- What goes wrong: You move terms and lose a minus.
- Why it’s wrong: One sign flip can change the entire solution set.
- Avoid it: Write each step; circle negatives; do a 5-second plug-back check.
Forgetting to flip inequalities when multiplying/dividing by a negative
- What goes wrong: You solve correctly but keep the same inequality direction.
- Why it’s wrong: Inequality direction reverses with negative scaling.
- Avoid it: Big reminder: “Negative? Flip it.”
Domain/denominator restrictions ignored
- What goes wrong: You cancel factors like $\frac{x-3}{x-3}$ and forget $x\ne 3$.
- Why it’s wrong: You change the function’s allowed inputs.
- Avoid it: Before simplifying rationals, note where denominators = 0.
Extraneous solutions after squaring
- What goes wrong: You square both sides to remove a root/absolute value and accept all solutions.
- Why it’s wrong: Squaring can introduce solutions that don’t satisfy the original.
- Avoid it: Always plug back if you squared or cleared absolute values.
Misreading “less than” language
- What goes wrong: “5 less than $x$” becomes $5-x$.
- Why it’s wrong: Word order matters.
- Avoid it: Translate carefully: “less than” usually means subtract from the following quantity: $x-5$.
Percent vs. percentage points
- What goes wrong: You treat a change from 40% to 50% as a 10% increase.
- Why it’s wrong: That’s +10 percentage points, but percent increase is $\frac{10}{40}=25\%$.
- Avoid it: Ask: “Is it an absolute difference, or relative to the original?”
Wrong radius/diameter in circle problems
- What goes wrong: They give diameter, you plug into $\pi r^2$ as if it’s $r$.
- Why it’s wrong: Area depends on $r^2$, so the error is huge.
- Avoid it: If you see $d$, immediately write $r=\frac{d}{2}$.
Using the calculator for everything (time sink)
- What goes wrong: You type long expressions instead of doing quick algebra.
- Why it’s wrong: Input errors + time loss.
- Avoid it: Use the calculator for graphs/intersections/verification—do simple operations by hand.
Not matching answer form (SPR especially)
- What goes wrong: You enter 0.3333 when exact $\frac{1}{3}$ is expected, or round too early.
- Why it’s wrong: Rounding can push you out of accepted tolerance.
- Avoid it: Keep exact fractions when possible; don’t round unless instructed.
Assuming diagrams are to scale
- What goes wrong: You “eyeball” angles/lengths.
- Why it’s wrong: SAT diagrams are not guaranteed to be to scale unless stated.
- Avoid it: Use given info + equations, not the picture.
6. Memory Aids & Mnemonics (include if applicable)
| Mnemonic | Stands for | When to use |
|---|---|---|
| PEMDAS | Parentheses, Exponents, Multiply/Divide, Add/Subtract | Order of operations (especially when simplifying expressions) |
| SOHCAHTOA | $\sin=\frac{\text{opp}}{\text{hyp}}$, $\cos=\frac{\text{adj}}{\text{hyp}}$, $\tan=\frac{\text{opp}}{\text{adj}}$ | If basic trig appears (rare but possible); right triangles only |
| “Rise over run” | Slope = $\frac{\Delta y}{\Delta x}$ | Quick slope from two points or a graph |
| Keep-Change-Flip | Keep first fraction, change ÷ to ×, flip second fraction | Dividing fractions/rational expressions |
| FOIL | First, Outer, Inner, Last | Multiply two binomials (when it actually helps) |
| “Negative? Flip it.” | Flip inequality sign | Solving inequalities with negative multiply/divide |
| “Vertex x is -b/2a” | x=-\frac{b}{2a}$$ | Fast vertex/axis of symmetry for quadratics |
7. Important Dates & Deadlines (include if applicable)
College Board updates SAT dates, registration deadlines, and score release timelines by testing year and region.
| Item | Where to verify (official) |
|---|---|
| Upcoming SAT test dates | College Board SAT Dates page (your region) |
| Registration + late registration deadlines/fees | Your College Board SAT registration dashboard |
| Score release date for your test | The score release info shown in your College Board account |
Don’t rely on social media screenshots for deadlines—log in and confirm.
8. Last-Minute Tips & Test Day Checklist
Night-before (math-focused)
- Review this list: linear forms, exponent rules, quadratic essentials, circle/triangle formulas.
- Do a 5-minute calculator warm-up: graph a line, find intersection, read a zero (x-intercept).
- Decide your defaults:
- If stuck: eliminate → guess → flag.
- If you square/clear denominators: check solutions.
What to bring
- Accepted device with Bluebook installed + charger (and any allowed accessories per your test center)
- Photo ID (and any required admission/registration info)
- Approved calculator (optional, since built-in exists) + fresh batteries if handheld
- Pencils/pens (even though it’s digital, you need them for scratch)
What NOT to bring / do
- Don’t depend on phone access (usually not allowed during testing; policies are strict).
- Don’t bring unapproved notes/formula sheets.
- Don’t start Module 2 mentally defeated—treat it as fresh points.
During the test (quick mental rules)
- Write equations first, then compute. The SAT rewards setup.
- If choices are ugly, backsolve or graph.
- After you solve, do a 10-second check: plug in, estimate, or verify units.
You’ve got this: if you keep your algebra clean and your checks consistent, you’ll pick up a ton of “free” points tomorrow.